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Contents

1 Solid harmonics 11.1 Derivation, relation to spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Racahs normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Real form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Linear combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 z-dependent part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.3 (x,y)-dependent part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.4 In total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.5 List of lowest functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Sonine formula 52.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Spherical harmonics 63.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Laplaces spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2.1 Orbital angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3.1 Orthogonality and normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.2 CondonShortley phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.3 Real form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4 Spherical harmonics in Cartesian form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.2 Real form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 Spherical harmonics expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.6 Spectrum analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.6.1 Power spectrum in signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6.2 Dierentiability properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.7 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7.1 Addition theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7.2 ClebschGordan coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

i

ii CONTENTS

3.7.3 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.8 Visualization of the spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.9 List of spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.10 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11 Connection with representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.11.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.12 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.13 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 StieltjesWigert polynomials 254.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Table of spherical harmonics 275.1 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1.1 l = 0[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1.2 l = 1[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1.3 l = 2[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1.4 l = 3[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1.5 l = 4[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1.6 l = 5[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1.7 l = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.1.8 l = 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.1.9 l = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.1.10 l = 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.11 l = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Real spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.1 l = 0[2][3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.2 l = 1[2][3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.3 l = 2[2][3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.4 l = 3[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2.5 l = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Toronto function 366.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7 Trigonometric integral 37

CONTENTS iii

7.1 Sine integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2 Cosine integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.3 Sinhc function integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.4 Hyperbolic cosine integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.5 Auxiliary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.6 Nielsens spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.7 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7.7.1 Asymptotic series (for large argument) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.7.2 Convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.8 Relation with the exponential integral of imaginary argument . . . . . . . . . . . . . . . . . . . . . 417.9 Ecient evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.10.1 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.13 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.13.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.13.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.13.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Chapter 1

Solid harmonics

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates.There are two kinds: the regular solid harmonicsRm` (r) , which vanish at the origin and the irregular solid harmonicsIm` (r) , which are singular at the origin. Both sets of functions play an important role in potential theory, and areobtained by rescaling spherical harmonics appropriately:

Rm` (r) r

4

2`+ 1r`Y m` (; ')

Im` (r) r

4

2`+ 1

Y m` (; ')

r`+1

1.1 Derivation, relation to spherical harmonicsIntroducing r, , and for the spherical polar coordinates of the 3-vector r, we can write the Laplace equation in thefollowing form

r2(r) = 1

r

@2

@r2r l^

2

r2

!(r) = 0; r 6= 0;

where l2 is the square of the nondimensional angular momentum operator,

l^ = i (rr):It is known that spherical harmonics Ym are eigenfunctions of l2:

l^2Y m` hl^2x + l^

2y + l^

2z

iY m` = `(`+ 1)Y

m` :

Substitution of (r) = F(r) Ym into the Laplace equation gives, after dividing out the spherical harmonic function,the following radial equation and its general solution,

1

r

@2

@r2rF (r) =

`(`+ 1)

r2F (r) =) F (r) = Ar` +Br`1:

The particular solutions of the total Laplace equation are regular solid harmonics:

Rm` (r) r

4

2`+ 1r`Y m` (; ');

1

2 CHAPTER 1. SOLID HARMONICS

and irregular solid harmonics:

Im` (r) r

4

2`+ 1

Y m` (; ')

r`+1:

1.1.1 Racahs normalization

Racah's normalization (also known as Schmidts semi-normalization) is applied to both functions

Z 0

sin dZ 20

d' Rm` (r) Rm` (r) =4

2`+ 1r2`

(and analogously for the irregular solid harmonic) instead of normalization to unity. This is convenient because inmany applications the Racah normalization factor appears unchanged throughout the derivations.

1.2 Addition theoremsThe translation of the regular solid harmonic gives a nite expansion,

Rm` (r+ a) =X`=0

2`

2

1/2 X=

R(r)Rm` (a) h; ; ` ;m j`mi;

where the Clebsch-Gordan coecient is given by

h; ; ` ;m j`mi =`+m

+

1/2`m

1/22`

2

1/2:

The similar expansion for irregular solid harmonics gives an innite series,

Im` (r+ a) =1X=0

2`+ 2+ 1

2

1/2 X=

R(r)Im`+ (a) h; ; `+ ;m j`mi

with jrj jaj . The quantity between pointed brackets is again a Clebsch-Gordan coecient,

h; ; `+ ;m j`mi = (1)+`+ m+

+

1/2`+ +m

1/2

2`+ 2+ 1

2

1/2:

1.2.1 References

The addition theorems were proved in dierent manners by many dierent workers. See for two dierent proofs forexample:

R. J. A. Tough and A. J. Stone, J. Phys. A: Math. Gen. Vol. 10, p. 1261 (1977)

M. J. Caola, J. Phys. A: Math. Gen. Vol. 11, p. L23 (1978)

1.3. REAL FORM 3

1.3 Real formBy a simple linear combination of solid harmonics of m these functions are transformed into real functions. Thereal regular solid harmonics, expressed in cartesian coordinates, are homogeneous polynomials of order l in x, y, z.The explicit form of these polynomials is of some importance. They appear, for example, in the form of sphericalatomic orbitals and real multipole moments. The explicit cartesian expression of the real regular harmonics will nowbe derived.

1.3.1 Linear combinationWe write in agreement with the earlier denition

Rm` (r; ; ') = (1)(m+jmj)/2 r` jmj` (cos )eim'; ` m `;

with

m` (cos ) (`m)!(`+m)!

1/2sinm d

mP`(cos )d cosm ; m 0;

where P`(cos ) is a Legendre polynomial of order l. Them dependent phase is known as the Condon-Shortley phase.The following expression denes the real regular solid harmonics:

Cm`Sm`

p2 r` m`

cosm'sinm'

=

1p2

(1)m 1(1)mi i

Rm`Rm`

; m > 0:

and for m = 0:

C0` R0` :

Since the transformation is by a unitary matrix the normalization of the real and the complex solid harmonics is thesame.

1.3.2 z-dependent partUpon writing u = cos the mth derivative of the Legendre polynomial can be written as the following expansion in u

dmP`(u)

dum=

b(`m)/2cXk=0

(m)`k u

`2km

with

(m)`k = (1)k2`

`

k

2` 2k

`

(` 2k)!

(` 2k m)! :

Since z = r cos it follows that this derivative, times an appropriate power of r, is a simple polynomial in z,

m` (z) r`mdmP`(u)

dum=

b(`m)/2cXk=0

(m)`k r

2k z`2km:

4 CHAPTER 1. SOLID HARMONICS

1.3.3 (x,y)-dependent partConsider next, recalling that x = r sincos and y = r sinsin,

rm sinm cosm' = 12

(r sin ei')m + (r sin ei')m

=

1

2[(x+ iy)m + (x iy)m]

Likewise

rm sinm sinm' = 12i

(r sin ei')m (r sin ei')m = 1

2i[(x+ iy)m (x iy)m] :

Further

Am(x; y) 12[(x+ iy)m + (x iy)m] =

mXp=0

m

p

xpymp cos(m p)

2

and

Bm(x; y) 12i

[(x+ iy)m (x iy)m] =mXp=0

m

p

xpymp sin(m p)

2:

1.3.4 In total

Cm` (x; y; z) =

(2 m0)(`m)!

(`+m)!

1/2m` (z) Am(x; y); m = 0; 1; : : : ; `

Sm` (x; y; z) =

2(`m)!(`+m)!

1/2m` (z) Bm(x; y); m = 1; 2; : : : ; `:

1.3.5 List of lowest functions

We list explicitly the lowest functions up to and including l = 5 . Here m` (z) h(2m0)(`m)!

(`+m)!

i1/2m` (z):

00 = 1 13 =

1

4

p6(5z2 r2) 44 =

1

8

p35

01 = z23 =

1

2

p15 z 05 =

1

8z(63z4 70z2r2 + 15r4)

11 = 1 33 =

1

4

p10 15 =

1

8

p15(21z4 14z2r2 + r4)

02 =1

2(3z2 r2) 04 =

1

8(35z4 30r2z2 + 3r4) 25 =

1

4

p105(3z2 r2)z

12 =p3z 14 =

p10

4z(7z2 3r2) 35 =

1

16

p70(9z2 r2)

22 =1

2

p3 24 =

1

4

p5(7z2 r2) 45 =

3

8

p35z

03 =1

2z(5z2 3r2) 34 =

1

4

p70 z 55 =

3

16

p14

The lowest functions Am(x; y) and Bm(x; y) are:

Chapter 2

Sonine formula

Inmathematics, Sonines formula is any of several formulas involving Bessel functions found byNikolay YakovlevichSonin.One such formula is the following integral formula involving a product of three Bessel functions:

Z 10

Jz(at)Jz(bt)Jz(ct)t1z dt =

2z1(a; b; c)2z1

1/2(z + 12 )(abc)z

where is the area of a triangle with given sides.

2.1 References Stempak, Krzysztof (1988), A new proof of Sonines formula, Proceedings of the American MathematicalSociety 104 (2): 453457, doi:10.2307/2046994, ISSN 0002-9939, MR 962812

5

Chapter 3

Spherical harmonics

Ylm redirects here. For other uses, see YLM (disambiguation).In mathematics, spherical harmonics are a series of special functions dened on the surface of a sphere used to

Visual representations of the rst few real spherical harmonics. Blue portions represent regions where the function is positive, andyellow portions represent where it is negative. The distance of the surface from the origin indicates the value of Y m` (; ) in angulardirection (; ) .

solve some kinds of dierential equations. As Fourier series are a series of functions used to represent functions ona circle, spherical harmonics are a series of functions that are used to represent functions dened on the surface ofa sphere. Spherical harmonics are functions dened in terms of spherical coordinates and are organized by angularfrequency, as seen in the rows of functions in the illustration on the right.Spherical harmonics are dened as the angular portion of a set of solutions to Laplaces equation in three dimensions.Represented in a system of spherical coordinates, Laplaces spherical harmonics Y m` are a specic set of sphericalharmonics that forms an orthogonal system, rst introduced by Pierre Simon de Laplace in 1782.[1]

Spherical harmonics are important in many theoretical and practical applications, particularly in the computationof atomic orbital electron congurations, representation of gravitational elds, geoids, and the magnetic elds ofplanetary bodies and stars, and characterization of the cosmic microwave background radiation. In 3D computergraphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion,

6

3.1. HISTORY 7

global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes.

3.1 HistorySpherical harmonics were rst investigated in connection with the Newtonian potential of Newtons law of universalgravitation in three dimensions. In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined thatthe gravitational potential at a point x associated to a set of point masses mi located at points xi was given by

V (x) =Xi

mijxi xj :

Each term in the above summation is an individual Newtonian potential for a point mass. Just prior to that time,Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = |x| and r1 = |x1|.He discovered that if r r1 then

1

jx1 xj = P0(cos )1

r1+ P1(cos )

r

r21+ P2(cos )

r2

r31+

where is the angle between the vectors x and x1. The functions Pi are the Legendre polynomials, and they area special case of spherical harmonics. Subsequently, in his 1782 memoire, Laplace investigated these coecientsusing spherical coordinates to represent the angle between x1 and x. (See Applications of Legendre polynomials inphysics for a more detailed analysis.)In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in theirTreatise on Natural Philosophy, and also rst introduced the name of spherical harmonics for these functions. Thesolid harmonics were homogeneous solutions of Laplaces equation

@2u

@x2+

@2u

@y2+

@2u

@z2= 0:

By examining Laplaces equation in spherical coordinates, Thomson and Tait recovered Laplaces spherical har-monics. The term Laplaces coecients was employed by William Whewell to describe the particular system ofsolutions introduced along these lines, whereas others reserved this designation for the zonal spherical harmonics thathad properly been introduced by Laplace and Legendre.The 19th century development of Fourier series made possible the solution of a wide variety of physical problems inrectangular domains, such as the solution of the heat equation and wave equation. This could be achieved by expansionof functions in series of trigonometric functions. Whereas the trigonometric functions in a Fourier series represent thefundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes of vibrationof a sphere in much the same way. Many aspects of the theory of Fourier series could be generalized by takingexpansions in spherical harmonics rather than trigonometric functions. This was a boon for problems possessingspherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre.The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th cen-tury birth of quantum mechanics. The spherical harmonics are eigenfunctions of the square of the orbital angularmomentum operator

i~rr;and therefore they represent the dierent quantized congurations of atomic orbitals.

3.2 Laplaces spherical harmonicsLaplaces equation imposes that the divergence of the gradient of a scalar eld f is zero. In spherical coordinates thisis:[2]

8 CHAPTER 3. SPHERICAL HARMONICS

Real (Laplace) spherical harmonics Ym for = 0, , 4 (top to bottom) and m = 0, , 4 (left to right). Zonal, sectoral, and tesseralharmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonicsY m` would be shown rotated about the z axis by 90/m with respect to the positive order ones.)

r2f = 1r2

@

@r

r2@f

@r

+

1

r2 sin @

@

sin @f

@

+

1

r2 sin2 @2f

@'2= 0:

Consider the problem of nding solutions of the form f(r, , ) = R(r) Y(, ). By separation of variables, twodierential equations result by imposing Laplaces equation:

1

R

d

dr

r2dR

dr

= ;

1

Y

1

sin @

@

sin @Y

@

+

1

Y

1

sin2 @2Y

@'2= :

The second equation can be simplied under the assumption that Y has the form Y(, ) = () (). Applyingseparation of variables again to the second equation gives way to the pair of dierential equations

1

d2

d'2= m2

3.2. LAPLACES SPHERICAL HARMONICS 9

sin2 + sin

d

d

sin d

d

= m2

for some number m. A priori, m is a complex constant, but because must be a periodic function whose periodevenly divides 2, m is necessarily an integer and is a linear combination of the complex exponentials e Im .The solution function Y(, ) is regular at the poles of the sphere, where = 0, . Imposing this regularity in thesolution of the second equation at the boundary points of the domain is a SturmLiouville problem that forces theparameter to be of the form = ( + 1) for some non-negative integer with |m|; this is also explained below interms of the orbital angular momentum. Furthermore, a change of variables t = cos transforms this equation intothe Legendre equation, whose solution is a multiple of the associated Legendre polynomial Pm(cos ) . Finally, theequation for R has solutions of the form R(r) = A r + B r 1; requiring the solution to be regular throughout R3forces B = 0.[3]

Here the solution was assumed to have the special form Y(, ) = () (). For a given value of , there are 2 +1 independent solutions of this form, one for each integer m with m . These angular solutions are a productof trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:

Y m` (; ') = Neim'Pm` (cos )

which fulll

r2r2Y m` (; ') = `(`+ 1)Y m` (; '):

Here Ym is called a spherical harmonic function of degree and orderm, Pm is an associated Legendre polynomial,N is a normalization constant, and and represent colatitude and longitude, respectively. In particular, the colatitude, or polar angle, ranges from 0 at the North Pole, to /2 at the Equator, to at the South Pole, and the longitude ,or azimuth, may assume all values with 0 < 2. For a xed integer , every solution Y(, ) of the eigenvalueproblem

r2r2Y = `(`+ 1)Y

is a linear combination of Ym. In fact, for any such solution, r Y(, ) is the expression in spherical coordinatesof a homogeneous polynomial that is harmonic (see below), and so counting dimensions shows that there are 2 + 1linearly independent such polynomials.The general solution to Laplaces equation in a ball centered at the origin is a linear combination of the sphericalharmonic functions multiplied by the appropriate scale factor r,

f(r; ; ') =1X`=0

X`m=`

fm` r`Y m` (; ');

where the fm are constants and the factors r Ym are known as solid harmonics. Such an expansion is valid in theball

r < R =1

lim sup`!1 jfm` j1`

:

3.2.1 Orbital angular momentumIn quantum mechanics, Laplaces spherical harmonics are understood in terms of the orbital angular momentum[4]

L = i~xr = Lxi+ Lyj+ Lzk:


The is conventional in quantummechanics; it is convenient to work in units in which = 1. The spherical harmonicsare eigenfunctions of the square of the orbital angular momentum

L2 = r2r2 +r@

@r+ 1

r@

@r

= 1sin @

@sin @

@ 1sin2

@2

@'2:

Laplaces spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and thegenerator of rotations about the azimuthal axis:

Lz = ix@

@y y @

@x

= i @

@':

These operators commute, and are densely dened self-adjoint operators on the Hilbert space of functions f square-integrable with respect to the normal distribution on R3:

1

(2)3/2

ZR3jf(x)j2ejxj2/2 dx

3.3. CONVENTIONS 11

Then, since

LL+ = L2 L2z Lzit follows that

0 = LLk+Y = ( (m+ k)2 (m+ k))Y:

Thus = (+1) for the positive integer = m+k.

3.3 Conventions

3.3.1 Orthogonality and normalizationSeveral dierent normalizations are in common use for the Laplace spherical harmonic functions. Throughout thesection, we use the standard convention that (see associated Legendre polynomials)

Pm` = (1)m(`m)!(`+m)!

Pm`

which is the natural normalization given by Rodrigues formula.In seismology, the Laplace spherical harmonics are generally dened as (this is the convention used in this article)

Y m` (; ') =

s(2`+ 1)

4

(`m)!(`+m)!

Pm` (cos ) eim'

while in quantum mechanics:[5][6]

Y m` (; ') = (1)ms

(2`+ 1)

4

(`m)!(`+m)!

Pm` (cos ) eim'

which are orthonormal

Z =0

Z 2'=0

Y m` Ym0`0

d = ``0 mm0 ;

where ij is the Kronecker delta and d = sin d d. This normalization is used in quantum mechanics because itensures that probability is normalized, i.e.

ZjY m` j2d = 1:

The disciplines of geodesy and spectral analysis use

Y m` (; ') =

s(2`+ 1)

(`m)!(`+m)!

Pm` (cos ) eim'

which possess unit power


1

4

Z =0

Z 2'=0

Y m` Ym0`0

d = ``0 mm0 :

The magnetics community, in contrast, uses Schmidt semi-normalized harmonics

Y m` (; ') =

s(`m)!(`+m)!

Pm` (cos ) eim'

which have the normalization

Z =0

Z 2'=0

Y m` Ym0`0

d =4

(2`+ 1)``0 mm0 :

In quantum mechanics this normalization is sometimes used as well, and is named Racahs normalization after GiulioRacah.It can be shown that all of the above normalized spherical harmonic functions satisfy

Y m`(; ') = (1)mY m` (; ');

where the superscript * denotes complex conjugation. Alternatively, this equation follows from the relation of thespherical harmonic functions with the Wigner D-matrix.

3.3.2 CondonShortley phaseOne source of confusion with the denition of the spherical harmonic functions concerns a phase factor of (1)m form > 0, 1 otherwise, commonly referred to as the CondonShortley phase in the quantum mechanical literature. Inthe quantum mechanics community, it is common practice to either include this phase factor in the denition of theassociated Legendre polynomials, or to append it to the denition of the spherical harmonic functions. There is norequirement to use the CondonShortley phase in the denition of the spherical harmonic functions, but includingit can simplify some quantum mechanical operations, especially the application of raising and lowering operators.The geodesy[7] and magnetics communities never include the CondonShortley phase factor in their denitions of thespherical harmonic functions nor in the ones of the associated Legendre polynomials.

3.3.3 Real formA real basis of spherical harmonics can be dened in terms of their complex analogues by setting

Y`m =

8>>>>>>>:ip2

Y m` (1)m Y m`

ifm < 0

Y 0` ifm = 01p2

Y m` + (1)m Y m`

ifm > 0:

=

8>>>>>>>:ip2

Yjmj` (1)m Y jmj`

ifm < 0

Y 0` ifm = 01p2

Yjmj` + (1)m Y jmj`

ifm > 0:

=

8>:p2 (1)m Im[Y jmj` ] ifm < 0

Y 0` ifm = 0p2 (1)m Re[Y m` ] ifm > 0:

3.4. SPHERICAL HARMONICS IN CARTESIAN FORM 13

The Condon-Shortley phase convention is used here for consistency. The corresponding inverse equations are

Y m` =

8>>>>>>>:1p2

Y`jmj iY`;jmj

ifm < 0

Y`0 ifm = 0(1)mp

2

Y`jmj + iY`;jmj

ifm > 0:

The real spherical harmonics are sometimes known as tesseral spherical harmonics.[8] These functions have the sameorthonormality properties as the complex ones above. The harmonics with m > 0 are said to be of cosine type, andthose with m < 0 of sine type. The reason for this can be seen by writing the functions in terms of the Legendrepolynomials as

Y`m =

8>>>>>>>>>>>>>>>:

p2

s(2`+ 1)

4

(` jmj)!(`+ jmj)!P

jmj` (cos ) sin jmj' ifm < 0r

(2`+ 1)

4Pm` (cos ) ifm = 0

p2

s(2`+ 1)

4

(`m)!(`+m)!

Pm` (cos ) cosm' ifm > 0

The same sine and cosine factors can be also seen in the following subsection that deals with the cartesian represen-tation.See here for a list of real spherical harmonics up to and including ` = 4 , which can be seen to be consistent with theoutput of the equations above.

Use in quantum chemistry

As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wavefunction are spherical harmonics. However, the solutions of the non-relativistic Schrdinger equation without mag-netic terms can be made real. This is why the real forms are extensively used in basis functions for quantum chemistry,as the programs don't then need to use complex algebra. Here, it is important to note that the real functions span thesame space as the complex ones would.For example, as can be seen from the table of spherical harmonics, the usual p functions ( l = 1 ) are complex andmix axis directions, but the real versions are essentially just x, y and z.

3.4 Spherical harmonics in Cartesian formThe following expresses normalized spherical harmonics in Cartesian coordinates (Condon-Shortley phase):

r`Y m`Y m`

=

2`+ 1

4

1/2m` (z)

(1)m(Am + iBm)

(Am iBm); m > 0:

and for m = 0:

r` Y 0` r

2`+ 1

40` :

Here

Am(x; y) =mXp=0

m

p

xpymp cos((m p)

2);


Bm(x; y) =mXp=0

m

p

xpymp sin((m p)

2);

and

m` (z) =

(`m)!(`+m)!

1/2 b(`m)/2cXk=0

(1)k2``

k

2` 2k

`

(` 2k)!

(` 2k m)! r2k z`2km:

Form = 0 this reduces to

0`(z) =

b`/2cXk=0

(1)k2``

k

2` 2k

`

r2k z`2k:

3.4.1 ExamplesUsing the expressions for `m(z) , Am(x; y) , and Bm(x; y) listed explicitly above we obtain:

Y 13 = 1

r374 316

1/2(5z2 r2)(x+ iy) = 74 3161/2 (5 cos2 1)(sin ei')

Y 24 =1

r494 532

1/2(7z2 r2)(x iy)2 = 94 5321/2 (7 cos2 1)(sin2 e2i')

It may be veried that this agrees with the function listed here and here.

3.4.2 Real formUsing the equations above to form the real spherical harmonics, it is seen that form > 0 only theAm terms (cosines)are included, and form < 0 only the Bm terms (sines) are included:

r`Y`mY`m

=

2`+ 1

4

1/2m` (z)

AmBm

; m > 0:

and for m = 0:

r` Y`0 r

2`+ 1

40` :

3.5 Spherical harmonics expansionThe Laplace spherical harmonics form a complete set of orthonormal functions and thus form an orthonormal basisof the Hilbert space of square-integrable functions. On the unit sphere, any square-integrable function can thus beexpanded as a linear combination of these:

f(; ') =

1X`=0

X`m=`

fm` Ym` (; '):

This expansion holds in the sense of mean-square convergence convergence in L2 of the sphere which is to saythat

3.6. SPECTRUM ANALYSIS 15

limN!1

Z 20

Z 0

f(; ')NX`=0

X`m=`

fm` Ym` (; ')

2

sin d d = 0:

The expansion coecients are the analogs of Fourier coecients, and can be obtained by multiplying the aboveequation by the complex conjugate of a spherical harmonic, integrating over the solid angle , and utilizing the aboveorthogonality relationships. This is justied rigorously by basic Hilbert space theory. For the case of orthonormalizedharmonics, this gives:

fm` =

Z

f(; ')Y m` (; ') d =Z 20

d'

Z 0

d sin f(; ')Y m` (; '):

If the coecients decay in suciently rapidly for instance, exponentially then the series also convergesuniformly to f.A square-integrable function f can also be expanded in terms of the real harmonics Ym above as a sum

f(; ') =1X`=0

X`m=`

f`m Y`m(; '):

The convergence of the series holds again in the same sense, but the benet of the real expansion is that for realfunctions f the expansion coecients become real.

3.6 Spectrum analysis

3.6.1 Power spectrum in signal processingThe total power of a function f is dened in the signal processing literature as the integral of the function squared,divided by the area of its domain. Using the orthonormality properties of the real unit-power spherical harmonicfunctions, it is straightforward to verify that the total power of a function dened on the unit sphere is related toits spectral coecients by a generalization of Parsevals theorem (here, the theorem is stated for Schmidt semi-normalized harmonics, the relationship is slightly dierent for orthonormal harmonics):

1

4

Z

jf()j2 d =1X`=0

Sff (`);

where

Sff (`) =1

2`+ 1

X`m=`

jf`mj2

is dened as the angular power spectrum (for Schmidt semi-normalized harmonics). In a similar manner, one candene the cross-power of two functions as

1

4

Z

f() g() d =1X`=0

Sfg(`);

where

Sfg(`) =1

2`+ 1

X`m=`

f`mg`m


is dened as the cross-power spectrum. If the functions f and g have a zero mean (i.e., the spectral coecients f00and g00 are zero), then S() and Sfg() represent the contributions to the functions variance and covariance fordegree , respectively. It is common that the (cross-)power spectrum is well approximated by a power law of theform

Sff (`) = C ` :

When = 0, the spectrum is white as each degree possesses equal power. When < 0, the spectrum is termedred as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when > 0, thespectrum is termed blue. The condition on the order of growth of S() is related to the order of dierentiabilityof f in the next section.

3.6.2 Dierentiability propertiesOne can also understand the dierentiability properties of the original function f in terms of the asymptotics of S().In particular, if S() decays faster than any rational function of as , then f is innitely dierentiable. If,furthermore, S() decays exponentially, then f is actually real analytic on the sphere.The general technique is to use the theory of Sobolev spaces. Statements relating the growth of the S() to dif-ferentiability are then similar to analogous results on the growth of the coecients of Fourier series. Specically,if

1X`=0

(1 + `2)sSff (`)

3.8. VISUALIZATION OF THE SPHERICAL HARMONICS 17

In particular, when x = y, this gives Unslds theorem[11]

X`m=`

Y `m(; ')Y`m(; ') =2`+ 1

4

which generalizes the identity cos2 + sin2 = 1 to two dimensions.In the expansion (1), the left-hand side P(xy) is a constant multiple of the degree zonal spherical harmonic.From this perspective, one has the following generalization to higher dimensions. Let Yj be an arbitrary orthonormalbasis of the space H of degree spherical harmonics on the n-sphere. Then Z(`)x , the degree zonal harmoniccorresponding to the unit vector x, decomposes as[12]

Furthermore, the zonal harmonic Z(`)x (y) is given as a constant multiple of the appropriate Gegenbauer polynomial:

Combining (2) and (3) gives (1) in dimension n = 2 when x and y are represented in spherical coordinates. Finally,evaluating at x = y gives the functional identity

dimH`!n1

=

dim(H`)Xj=1

jYj(x)j2

where n is the volume of the (n1)-sphere.

3.7.2 ClebschGordan coecients

Main article: ClebschGordan coecients

The ClebschGordan coecients are the coecients appearing in the expansion of the product of two sphericalharmonics in terms of spherical harmonics itself. A variety of techniques are available for doing essentially thesame calculation, including the Wigner 3-jm symbol, the Racah coecients, and the Slater integrals. Abstractly, theClebschGordan coecients express the tensor product of two irreducible representations of the rotation group as asum of irreducible representations: suitably normalized, the coecients are then the multiplicities.

3.7.3 Parity

Main article: Parity (physics)

The spherical harmonics have well dened parity in the sense that they are either even or odd with respect to reectionabout the origin. Reection about the origin is represented by the operator P(~r) = (~r) . For the sphericalangles, f; g this corresponds to the replacement f ; + g . The associated Legendre polynomials gives(1)+m and from the exponential function we have (1)m, giving together for the spherical harmonics a parity of(1):

Y m` (; )! Y m` ( ; + ) = (1)`Y m` (; )

This remains true for spherical harmonics in higher dimensions: applying a point reection to a spherical harmonicof degree changes the sign by a factor of (1).


Schematic representation of Y`m on the unit sphere and its nodal lines. Re[Y`m] is equal to 0 along m great circles passing throughthe poles, and along m circles of equal latitude. The function changes sign each time it crosses one of these lines.

3.8 Visualization of the spherical harmonics

The Laplace spherical harmonics Y m` can be visualized by considering their "nodal lines", that is, the set of pointson the sphere where Re[Y m` ] = 0 , or alternatively where Im[Y m` ] = 0 . Nodal lines of Y m` are composed of circles:some are latitudes and others are longitudes. One can determine the number of nodal lines of each type by countingthe number of zeros of Y m` in the latitudinal and longitudinal directions independently. For the latitudinal direction,the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, whereas for thelongitudinal direction, the trigonometric sin and cos functions possess 2|m| zeros.When the spherical harmonic orderm is zero (upper-left in the gure), the spherical harmonic functions do not dependupon longitude, and are referred to as zonal. Such spherical harmonics are a special case of zonal spherical functions.When = |m| (bottom-right in the gure), there are no zero crossings in latitude, and the functions are referred to assectoral. For the other cases, the functions checker the sphere, and they are referred to as tesseral.More general spherical harmonics of degree are not necessarily those of the Laplace basis Y m` , and their nodal setscan be of a fairly general kind.[13]

3.9. LIST OF SPHERICAL HARMONICS 19

3.9 List of spherical harmonicsMain article: Table of spherical harmonics

Analytic expressions for the rst few orthonormalized Laplace spherical harmonics that use the Condon-Shortleyphase convention:

Y 00 (; ') =1

2

r1

Y 11 (; ') =1

2

r3

2sin ei'

Y 01 (; ') =1

2

r3

cos

Y 11 (; ') =12

r3

2sin ei'

Y 22 (; ') =1

4

r15

2sin2 e2i'

Y 12 (; ') =1

2

r15

2sin cos ei'

Y 02 (; ') =1

4

r5

(3 cos2 1)

Y 12 (; ') =12

r15

2sin cos ei'

Y 22 (; ') =1

4

r15

2sin2 e2i'

3.10 Higher dimensionsThe classical spherical harmonics are dened as functions on the unit sphere S2 inside three-dimensional Euclideanspace. Spherical harmonics can be generalized to higher-dimensional Euclidean spaceRn as follows.[14] LetP denotethe space of homogeneous polynomials of degree in n variables. That is, a polynomial P is in P provided that

P (x) = `P (x):

Let A denote the subspace of P consisting of all harmonic polynomials; these are the solid spherical harmonics.Let H denote the space of functions on the unit sphere

Sn1 = fx 2 Rn j jxj = 1gobtained by restriction from A.The following properties hold:

The sum of the spaces H is dense in the set of continuous functions on Sn1 with respect to the uniformtopology, by the Stone-Weierstrass theorem. As a result, the sum of these spaces is also dense in the spaceL2(Sn1) of square-integrable functions on the sphere. Thus every square-integrable function on the spheredecomposes uniquely into a series a spherical harmonics, where the series converges in the L2 sense.

For all f H, one has


Sn1f = `(`+ n 2)f:

where Sn is the LaplaceBeltrami operator on Sn1. This operator is the analog of the angular part ofthe Laplacian in three dimensions; to wit, the Laplacian in n dimensions decomposes as

r2 = r1n @@r

rn1@

@r+ r2Sn1 :

It follows from the Stokes theorem and the preceding property that the spaces H are orthogonal with respectto the inner product from L2(Sn1). That is to say,

ZSn1

fg d = 0

for f H and g Hk for k .

Conversely, the spaces H are precisely the eigenspaces of Sn. In particular, an application of the spectraltheorem to the Riesz potential 1Sn1 gives another proof that the spaces H are pairwise orthogonal andcomplete in L2(Sn1).

Every homogeneous polynomial P P can be uniquely written in the form

P (x) = P`(x) + jxj2P`2 + +(jxj`P0 ` evenjxj`1P1(x) ` odd

where Pj Aj. In particular,

dimH` =n+ ` 1n 1

n+ ` 3n 1

:

An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method ofseparation of variables, by solving the Sturm-Liouville problem for the spherical Laplacian

Sn1 = sin2n @

@sinn2 @

@+ sin2 Sn2

where is the axial coordinate in a spherical coordinate system on Sn1. The end result of such a procedure is[15]

Yl1;:::ln1(1; : : : n1) =1p2

eil11n1Yj=2

jPln2lj

(j)

where the indices satisfy |1| 2 ... n and the eigenvalue is n(n + n2). The functions in the productare dened in terms of the Legendre function

jP lL() =

s2L+ j 1

2

(L+ l + j 2)!(L l)! sin

2j2 ()P

(l+ j22 )L+ j22

(cos )

3.11. CONNECTION WITH REPRESENTATION THEORY 21

3.11 Connection with representation theoryThe space H of spherical harmonics of degree is a representation of the symmetry group of rotations around apoint (SO(3)) and its double-cover SU(2). Indeed, rotations act on the two-dimensional sphere, and thus also on Hby function composition

7! for a spherical harmonic and a rotation. The representation H is an irreducible representation of SO(3).The elements of H arise as the restrictions to the sphere of elements of A: harmonic polynomials homogeneousof degree on three-dimensional Euclidean space R3. By polarization of A, there are coecients i1:::i`symmetric on the indices, uniquely determined by the requirement

(x1; : : : ; xn) =Xi1:::i`

i1:::i`xi1 xi` :

The condition that be harmonic is equivalent to the assertion that the tensor i1:::i` must be trace free on everypair of indices. Thus as an irreducible representation of SO(3),H is isomorphic to the space of traceless symmetrictensors of degree .More generally, the analogous statements hold in higher dimensions: the space H of spherical harmonics on then-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric -tensors. However,whereas every irreducible tensor representation of SO(2) and SO(3) is of this kind, the special orthogonal groups inhigher dimensions have additional irreducible representations that do not arise in this manner.The special orthogonal groups have additional spin representations that are not tensor representations, and are typicallynot spherical harmonics. An exception are the spin representation of SO(3): strictly speaking these are representationsof the double cover SU(2) of SO(3). In turn, SU(2) is identied with the group of unit quaternions, and so coincideswith the 3-sphere. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), withrespect to the action by quaternionic multiplication.

3.11.1 GeneralizationsThe angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C).With respect to this group, the sphere is equivalent to the usual Riemann sphere. The group PSL(2,C) is isomorphicto the (proper) Lorentz group, and its action on the two-sphere agrees with the action of the Lorentz group on thecelestial sphere in Minkowski space. The analog of the spherical harmonics for the Lorentz group is given by thehypergeometric series; furthermore, the spherical harmonics can be re-expressed in terms of the hypergeometricseries, as SO(3) = PSU(2) is a subgroup of PSL(2,C).More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; inparticular, hypergeometric series can be developed for any Lie group.[16][17][18][19]

3.12 See also Cylindrical harmonics Spherical basis Spin spherical harmonics Spin-weighted spherical harmonics SturmLiouville theory Table of spherical harmonics Vector spherical harmonics


3.13 Notes[1] A historical account of various approaches to spherical harmonics in three-dimensions can be found in Chapter IV of

MacRobert 1967. The term Laplace spherical harmonics is in common use; see Courant & Hilbert 1962 and Meijer &Bauer 2004.

[2] The approach to spherical harmonics taken here is found in (Courant & Hilbert 1966, V.8, VII.5).

[3] Physical applications often take the solution that vanishes at innity, making A = 0. This does not aect the angular portionof the spherical harmonics.

[4] Edmonds 1957, 2.5

[5] Messiah, Albert (1999). Quantum mechanics : two volumes bound as one (Two vol. bound as one, unabridged reprint ed.).Mineola, NY: Dover. ISBN 9780486409245.

[6] al.], Claude Cohen-Tannoudji, Bernard Diu, Franck Lalo; transl. from the French by Susan Reid Hemley ... [et (1996).Quantum mechanics. Wiley-Interscience: Wiley. ISBN 9780471569527.

[7] Heiskanen and Moritz, Physical Geodesy, 1967, eq. 1-62

[8] Watson & Whittaker 1927, p. 392.

[9] This is valid for any orthonormal basis of spherical harmonics of degree . For unit power harmonics it is necessary toremove the factor of 4.

[10] Watson & Whittaker 1927, p. 395

[11] Unsld 1927

[12] Stein & Weiss 1971, IV.2

[13] Eremenko, Jakobson & Nadirashvili 2007

[14] Solomentsev 2001; Stein & Weiss 1971, Iv.2

[15] Higuchi, Atsushi (1987). Symmetric tensor spherical harmonics on the N-sphere and their application to the de Sittergroup SO(N,1)". Journal of Mathematical Physics 28 (7). doi:10.1063/1.527513.

[16] N. Vilenkin, Special Functions and the Theory of Group Representations, Am. Math. Soc. Transl.,vol. 22, (1968).

[17] J. D. Talman, Special Functions, A Group Theoretic Approach, (based on lectures by E.P. Wigner), W. A. Benjamin, NewYork (1968).

[18] W. Miller, Symmetry and Separation of Variables, Addison-Wesley, Reading (1977).

[19] A. Wawrzyczyk, Group Representations and Special Functions, Polish Scientic Publishers. Warszawa (1984).

3.14 ReferencesCited references

Courant, Richard; Hilbert, David (1962), Methods of Mathematical Physics, Volume I, Wiley-Interscience. Edmonds, A.R. (1957), Angular Momentum in Quantum Mechanics, Princeton University Press, ISBN 0-691-07912-9

Eremenko, Alexandre; Jakobson, Dmitry; Nadirashvili, Nikolai (2007), On nodal sets and nodal domains onS and R", Universit de Grenoble. Annales de l'Institut Fourier 57 (7): 23452360, doi:10.5802/aif.2335,ISSN 0373-0956, MR 2394544

MacRobert, T.M. (1967), Spherical harmonics: An elementary treatise on harmonic functions, with applications,Pergamon Press.

Meijer, Paul Herman Ernst; Bauer, Edmond (2004), Group theory: The application to quantum mechanics,Dover, ISBN 978-0-486-43798-9.

3.14. REFERENCES 23

Solomentsev, E.D. (2001), Spherical harmonics, in Hazewinkel, Michiel, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4.

Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.:Princeton University Press, ISBN 978-0-691-08078-9.

Unsld, Albrecht (1927), Beitrge zur Quantenmechanik der Atome, Annalen der Physik 387 (3): 355393,Bibcode:1927AnP...387..355U, doi:10.1002/andp.19273870304.

Watson, G. N.; Whittaker, E. T. (1927), A Course of Modern Analysis, Cambridge University Press, p. 392.

General references

E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, (1955) Chelsea Pub. Co., ISBN 978-0-8284-0104-3.

C. Mller, Spherical Harmonics, (1966) Springer, Lecture Notes in Mathematics, Vol. 17, ISBN 978-3-540-03600-5.

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, (1970) Cambridge at the University Press,ISBN 0-521-09209-4, See chapter 3.

J.D. Jackson, Classical Electrodynamics, ISBN 0-471-30932-X Albert Messiah, Quantum Mechanics, volume II. (2000) Dover. ISBN 0-486-40924-4. Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), Section 6.7. Spherical Harmonics, Nu-merical Recipes: The Art of Scientic Computing (3rd ed.), New York: Cambridge University Press, ISBN978-0-521-88068-8

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii Quantum Theory of Angular Momentum,(1988)WorldScientic Publishing Co., Singapore, ISBN 9971-5-0107-4

Weisstein, Eric W., Spherical harmonics, MathWorld.


3D color plot of the spherical harmonics of degree n = 5. Note that n = .

Chapter 4

StieltjesWigert polynomials

Not to be confused with Stieltjes polynomial.For the generalized StieltjesWigert polynomials, see q-Laguerre polynomials.

In mathematics, StieltjesWigert polynomials (named after Thomas Jan Stieltjes and Carl Severin Wigert) are afamily of basic hypergeometric orthogonal polynomials in the basic Askey scheme, for the weight function [1]

w(x) =kpx1/2 exp(k2 log2 x)

on the positive real line x > 0.The moment problem for the StieltjesWigert polynomials is indeterminate; in other words, there are many othermeasures giving the same family of orthogonal polynomials (see Kreins condition).Koekoek et al. (2010) give in Section 14.27 a detailed list of the properties of these polynomials.

4.1 DenitionThe polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by[2]

Sn(x; q) =1

(q; q)n)11(q

n; 0; q;qn+1x)

(where q = e1 k2 ).

4.2 OrthogonalitySince the moment problem for these polynomials is indeterminate there are many dierent weight functions on [0,]for which they are orthogonal. Two examples of such weight functions are

1

(x;qx1; q)1and

kpx1/2 exp(k2 log2 x)

25

26 CHAPTER 4. STIELTJESWIGERT POLYNOMIALS

4.3 Notes[1] Up to a constant factor this is w(q1/2x) for the weight function w in Szeg (1975), Section 2.7. See also Koornwinder et

al. (2010), Section 18.27(vi).

[2] Up to a constant factor Sn(x;q)=pn(q1/2x) for pn(x) in Szeg (1975), Section 2.7.

4.4 References Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and itsApplications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8,MR 2128719

Koekoek, Roelof; Lesky, Peter A.; Swarttouw, Ren F. (2010), Hypergeometric orthogonal polynomials andtheir q-analogues, SpringerMonographs inMathematics, Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096

Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), Ch. 18, Or-thogonal polynomials, in Olver, FrankW. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NISTHandbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248

Szeg, Gbor (1975), Orthogonal Polynomials, Colloquium Publications 23, American Mathematical Society,Fourth Edition, ISBN 978-0-8218-1023-1, MR 0372517

Stieltjes, T. -J. (1894), Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse (in French) VIII:1122, JFM 25.0326.01, MR 1344720

Wigert, S. (1923), Sur les polynomes orthogonaux et l'approximation des fonctions continues, Arkiv frmatematik, astronomi och fysik (in French) 17: 115, JFM 49.0296.01

Chapter 5

Table of spherical harmonics

This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree l =10. Some of these formulas give the Cartesian version. This assumes x, y, z, and r are related to and ' throughthe usual spherical-to-Cartesian coordinate transformation:

x = r sin cos'y = r sin sin'z = r cos

5.1 Spherical harmonics

5.1.1 l = 0[1]

Y 00 (; ') =1

2

r1

5.1.2 l = 1[1]

Y 11 (; ') =1

2

r3

2 ei' sin = 1

2

r3

2 (x iy)

r

Y 01 (; ') =1

2

r3

cos = 1

2

r3

zr

Y 11 (; ') =12

r3

2 ei' sin = 1

2

r3

2 (x+ iy)

r

5.1.3 l = 2[1]

Y 22 (; ') =1

4

r15

2 e2i' sin2 = 1

4

r15

2 (x iy)

2

r2

Y 12 (; ') =1

2

r15

2 ei' sin cos = 1

2

r15

2 (x iy)z

r2

Y 02 (; ') =1

4

r5

(3 cos2 1) = 1

4

r5

(2z

2 x2 y2)r2

Y 12 (; ') =12

r15

2 ei' sin cos = 1

2

r15

2 (x+ iy)z

r2

Y 22 (; ') =1

4

r15

2 e2i' sin2 = 1

4

r15

2 (x+ iy)

2

r2

27

28 CHAPTER 5. TABLE OF SPHERICAL HARMONICS

5.1.4 l = 3[1]

Y 33 (; ') =1

8

r35

e3i' sin3 = 1

8

r35

(x iy)

3

r3

Y 23 (; ') =1

4

r105

2 e2i' sin2 cos = 1

4

r105

2 (x iy)

2z

r3

Y 13 (; ') =1

8

r21

ei' sin (5 cos2 1) = 1

8

r21

(x iy)(4z

2 x2 y2)r3

Y 03 (; ') =1

4

r7

(5 cos3 3 cos ) = 1

4

r7

z(2z

2 3x2 3y2)r3

Y 13 (; ') =18

r21

ei' sin (5 cos2 1) = 1

8

r21

(x+ iy)(4z

2 x2 y2)r3

Y 23 (; ') =1

4

r105

2 e2i' sin2 cos = 1

4

r105

2 (x+ iy)

2z

r3

Y 33 (; ') =18

r35

e3i' sin3 = 1

8

r35

(x+ iy)

3

r3

5.1.5 l = 4[1]

Y 44 (; ') =3

16

r35

2 e4i' sin4 = 3

16

r35

2 (x iy)

4

r4

Y 34 (; ') =3

8

r35

e3i' sin3 cos = 3

8

r35

(x iy)

3z

r4

Y 24 (; ') =3

8

r5

2 e2i' sin2 (7 cos2 1) = 3

8

r5

2 (x iy)

2 (7z2 r2)r4

Y 14 (; ') =3

8

r5

ei' sin (7 cos3 3 cos ) = 3

8

r5

(x iy) z (7z

2 3r2)r4

Y 04 (; ') =3

16

r1

(35 cos4 30 cos2 + 3) = 3

16

r1

(35z

4 30z2r2 + 3r4)r4

Y 14 (; ') =38

r5

ei' sin (7 cos3 3 cos ) = 3

8

r5

(x+ iy) z (7z

2 3r2)r4

Y 24 (; ') =3

8

r5

2 e2i' sin2 (7 cos2 1) = 3

8

r5

2 (x+ iy)

2 (7z2 r2)r4

Y 34 (; ') =38

r35

e3i' sin3 cos = 3

8

r35

(x+ iy)

3z

r4

Y 44 (; ') =3

16

r35

2 e4i' sin4 = 3

16

r35

2 (x+ iy)

4

r4

5.1.6 l = 5[1]

Y 55 (; ') =3

32

r77

e5i' sin5

Y 45 (; ') =3

16

r385

2 e4i' sin4 cos

Y 35 (; ') =1

32

r385

e3i' sin3 (9 cos2 1)

Y 25 (; ') =1

8

r1155

2 e2i' sin2 (3 cos3 cos )

5.1. SPHERICAL HARMONICS 29

Y 15 (; ') =1

16

r165

2 ei' sin (21 cos4 14 cos2 + 1)

Y 05 (; ') =1

16

r11

(63 cos5 70 cos3 + 15 cos )

Y 15 (; ') =116

r165

2 ei' sin (21 cos4 14 cos2 + 1)

Y 25 (; ') =1

8

r1155


Y 35 (; ') =132

r385

e3i' sin3 (9 cos2 1)

Y 45 (; ') =3

16

r385

2 e4i' sin4 cos

Y 55 (; ') =332

r77

e5i' sin5

5.1.7 l = 6

Y 66 (; ') =1

64

r3003

e6i' sin6

Y 56 (; ') =3

32

r1001

e5i' sin5 cos

Y 46 (; ') =3

32

r91

2 e4i' sin4 (11 cos2 1)

Y 36 (; ') =1

32

r1365

e3i' sin3 (11 cos3 3 cos )

Y 26 (; ') =1

64

r1365

e2i' sin2 (33 cos4 18 cos2 + 1)

Y 16 (; ') =1

16

r273

2 ei' sin (33 cos5 30 cos3 + 5 cos )

Y 06 (; ') =1

32

r13

(231 cos6 315 cos4 + 105 cos2 5)

Y 16 (; ') =116

r273

2 ei' sin (33 cos5 30 cos3 + 5 cos )

Y 26 (; ') =1

64

r1365

e2i' sin2 (33 cos4 18 cos2 + 1)

Y 36 (; ') =132

r1365


Y 46 (; ') =3

32

r91

2 e4i' sin4 (11 cos2 1)

Y 56 (; ') =332

r1001

e5i' sin5 cos

Y 66 (; ') =1

64

r3003

e6i' sin6


5.1.8 l = 7

Y 77 (; ') =3

64

r715

2 e7i' sin7

Y 67 (; ') =3

64

r5005

e6i' sin6 cos

Y 57 (; ') =3

64

r385

2 e5i' sin5 (13 cos2 1)

Y 47 (; ') =3

32

r385

2 e4i' sin4 (13 cos3 3 cos )

Y 37 (; ') =3

64

r35

2 e3i' sin3 (143 cos4 66 cos2 + 3)

Y 27 (; ') =3

64

r35

e2i' sin2 (143 cos5 110 cos3 + 15 cos )

Y 17 (; ') =1

64

r105

2 ei' sin (429 cos6 495 cos4 + 135 cos2 5)

Y 07 (; ') =1

32

r15

(429 cos7 693 cos5 + 315 cos3 35 cos )

Y 17 (; ') =164

r105

2 ei' sin (429 cos6 495 cos4 + 135 cos2 5)

Y 27 (; ') =3

64

r35

e2i' sin2 (143 cos5 110 cos3 + 15 cos )

Y 37 (; ') =364

r35

2 e3i' sin3 (143 cos4 66 cos2 + 3)

Y 47 (; ') =3

32

r385

2 e4i' sin4 (13 cos3 3 cos )

Y 57 (; ') =364

r385

2 e5i' sin5 (13 cos2 1)

Y 67 (; ') =3

64

r5005

e6i' sin6 cos

Y 77 (; ') =364

r715

2 e7i' sin7

5.1.9 l = 8

Y 88 (; ') =3

256

r12155

2 e8i' sin8

Y 78 (; ') =3

64

r12155

2 e7i' sin7 cos

Y 68 (; ') =1

128

r7293

e6i' sin6 (15 cos2 1)

Y 58 (; ') =3

64

r17017


Y 48 (; ') =3

128

r1309

2 e4i' sin4 (65 cos4 26 cos2 + 1)

Y 38 (; ') =1

64

r19635

2 e3i' sin3 (39 cos5 26 cos3 + 3 cos )

5.1. SPHERICAL HARMONICS 31

Y 28 (; ') =3

128

r595

e2i' sin2 (143 cos6 143 cos4 + 33 cos2 1)

Y 18 (; ') =3

64

r17

2 ei' sin (715 cos7 1001 cos5 + 385 cos3 35 cos )

Y 08 (; ') =1

256

r17

(6435 cos8 12012 cos6 + 6930 cos4 1260 cos2 + 35)

Y 18 (; ') =364

r17

2 ei' sin (715 cos7 1001 cos5 + 385 cos3 35 cos )

Y 28 (; ') =3

128

r595

e2i' sin2 (143 cos6 143 cos4 + 33 cos2 1)

Y 38 (; ') =164

r19635

2 e3i' sin3 (39 cos5 26 cos3 + 3 cos )

Y 48 (; ') =3

128

r1309

2 e4i' sin4 (65 cos4 26 cos2 + 1)

Y 58 (; ') =364

r17017


Y 68 (; ') =1

128

r7293

e6i' sin6 (15 cos2 1)

Y 78 (; ') =364

r12155

2 e7i' sin7 cos

Y 88 (; ') =3

256

r12155

2 e8i' sin8

5.1.10 l = 9

Y 99 (; ') =1

512

r230945

e9i' sin9

Y 89 (; ') =3

256

r230945

2 e8i' sin8 cos

Y 79 (; ') =3

512

r13585

e7i' sin7 (17 cos2 1)

Y 69 (; ') =1

128

r40755


Y 59 (; ') =3

256

r2717

e5i' sin5 (85 cos4 30 cos2 + 1)

Y 49 (; ') =3

128

r95095

2 e4i' sin4 (17 cos5 10 cos3 + cos )

Y 39 (; ') =1

256

r21945

e3i' sin3 (221 cos6 195 cos4 + 39 cos2 1)

Y 29 (; ') =3

128

r1045

e2i' sin2 (221 cos7 273 cos5 + 91 cos3 7 cos )

Y 19 (; ') =3

256

r95

2 ei' sin (2431 cos8 4004 cos6 + 2002 cos4 308 cos2 + 7)

Y 09 (; ') =1

256

r19

(12155 cos9 25740 cos7 + 18018 cos5 4620 cos3 + 315 cos )


Y 19 (; ') =3256

r95

2 ei' sin (2431 cos8 4004 cos6 + 2002 cos4 308 cos2 + 7)

Y 29 (; ') =3

128

r1045


Y 39 (; ') =1256

r21945

e3i' sin3 (221 cos6 195 cos4 + 39 cos2 1)

Y 49 (; ') =3

128

r95095

2 e4i' sin4 (17 cos5 10 cos3 + cos )

Y 59 (; ') =3256

r2717

e5i' sin5 (85 cos4 30 cos2 + 1)

Y 69 (; ') =1

128

r40755


Y 79 (; ') =3512

r13585

e7i' sin7 (17 cos2 1)

Y 89 (; ') =3

256

r230945

2 e8i' sin8 cos

Y 99 (; ') =1512

r230945

e9i' sin9

5.1.11 l = 10

Y 1010 (; ') =1

1024

r969969

e10i' sin10

Y 910 (; ') =1

512

r4849845

e9i' sin9 cos

Y 810 (; ') =1

512

r255255

2 e8i' sin8 (19 cos2 1)

Y 710 (; ') =3

512

r85085


Y 610 (; ') =3

1024

r5005

e6i' sin6 (323 cos4 102 cos2 + 3)

Y 510 (; ') =3

256

r1001

e5i' sin5 (323 cos5 170 cos3 + 15 cos )

Y 410 (; ') =3

256

r5005

2 e4i' sin4 (323 cos6 255 cos4 + 45 cos2 1)

Y 310 (; ') =3

256

r5005


Y 210 (; ') =3

512

r385

2 e2i' sin2 (4199 cos8 6188 cos6 + 2730 cos4 364 cos2 + 7)

Y 110 (; ') =1

256

r1155

2 ei' sin (4199 cos9 7956 cos7 + 4914 cos5 1092 cos3 + 63 cos )

Y 010(; ') =1

512

r21

(46189 cos10 109395 cos8 + 90090 cos6 30030 cos4 + 3465 cos2 63)

Y 110(; ') =1256

r1155

2 ei' sin (4199 cos9 7956 cos7 + 4914 cos5 1092 cos3 + 63 cos )

5.2. REAL SPHERICAL HARMONICS 33

Y 210(; ') =3

512

r385

2 e2i' sin2 (4199 cos8 6188 cos6 + 2730 cos4 364 cos2 + 7)

Y 310(; ') =3256

r5005


Y 410(; ') =3

256

r5005

2 e4i' sin4 (323 cos6 255 cos4 + 45 cos2 1)

Y 510(; ') =3256

r1001

e5i' sin5 (323 cos5 170 cos3 + 15 cos )

Y 610(; ') =3

1024

r5005

e6i' sin6 (323 cos4 102 cos2 + 3)

Y 710(; ') =3512

r85085


Y 810(; ') =1

512

r255255

2 e8i' sin8 (19 cos2 1)

Y 910(; ') =1512

r4849845

e9i' sin9 cos

Y 1010 (; ') =1

1024

r969969

e10i' sin10

5.2 Real spherical harmonicsFor each real spherical harmonic, the corresponding atomic orbital symbol (s, p, d, f, g) is reported as well.

5.2.1 l = 0[2][3]

Y00 = s = Y00 =

1

2

r1

5.2.2 l = 1[2][3]

Y1;1 = py = i

r1

2

Y 11 + Y

11

=

r3

4 yr

Y10 = pz = Y01 =

r3

4 zr

Y11 = px =

r1

2

Y 11 Y 11

=

r3

4 xr

5.2.3 l = 2[2][3]

Y2;2 = dxy = i

r1

2

Y 22 Y 22

=

1

2

r15

xyr2

Y2;1 = dyz = i

r1

2

Y 12 + Y

12

=

1

2

r15

yzr2

Y20 = dz2 = Y02 =

1

4

r5

x

2 y2 + 2z2r2

Y21 = dxz =

r1

2

Y 12 Y 12

=

1

2

r15

zxr2

Y22 = dx2y2 =

r1

2

Y 22 + Y

22

=

1

4

r15

x

2 y2r2


5.2.4 l = 3[2]

Y3;3 = fy(3x2y2) = i

r1

2

Y 33 + Y

33

=

1

4

r35

23x2 y2 y

r3

Y3;2 = fxyz = i

r1

2

Y 23 Y 23

=

1

2

r105

xyzr3

Y3;1 = fyz2 = i

r1

2

Y 13 + Y

13

=

1

4

r21

2 y(4z

2 x2 y2)r3

Y30 = fz3 = Y03 =

1

4

r7

z(2z

2 3x2 3y2)r3

Y31 = fxz2 =

r1

2

Y 13 Y 13

=

1

4

r21

2 x(4z

2 x2 y2)r3

Y32 = fz(x2y2) =

r1

2

Y 23 + Y

23

=

1

4

r105

x2 y2 z

r3

Y33 = fx(x23y2) =

r1

2

Y 33 Y 33

=

1

4

r35

2x2 3y2x

r3

5.2.5 l = 4

Y4;4 = gxy(x2y2) = i

r1

2

Y 44 Y 44

=

3

4

r35

xy

x2 y2r4

Y4;3 = gzy3 = i

r1

2

Y 34 + Y

34

=

3

4

r35

2 (3x

2 y2)yzr4

Y4;2 = gz2xy = i

r1

2

Y 24 Y 24

=

3

4

r5

xy (7z

2 r2)r4

Y4;1 = gz3y = i

r1

2

Y 14 + Y

14

=

3

4

r5

2 yz (7z

2 3r2)r4

Y40 = gz4 = Y04 =

3

16

r1

(35z

4 30z2r2 + 3r4)r4

Y41 = gz3x =

r1

2

Y 14 Y 14

=

3

4

r5

2 xz (7z

2 3r2)r4

Y42 = gz2xy =

r1

2

Y 24 + Y

24

=

3

8

r5

(x

2 y2) (7z2 r2)r4

Y43 = gzx3 =

r1

2

Y 34 Y 34

=

3

4

r35

2 (x

2 3y2)xzr4

Y44 = gx4+y4 =

r1

2

Y 44 + Y

44

=

3

16

r35

x

2x2 3y2 y2 3x2 y2

r4

5.3 See also Spherical harmonics

5.4 External links Spherical Harmonic at MathWorld

5.5 ReferencesCited references

5.5. REFERENCES 35

[1] D. A. Varshalovich, A. N.Moskalev, V. K. Khersonskii (1988). Quantum theory of angular momentum : irreducible tensors,spherical harmonics, vector coupling coecients, 3nj symbols (1. repr. ed.). Singapore: World Scientic Pub. p. 155-156.ISBN 9971-50-107-4.

[2] C.D.H. Chisholm (1976). Group theoretical techniques in quantum chemistry. New York: Academic Press. ISBN 0-12-172950-8.

[3] Blanco, Miguel A.; Flrez, M.; Bermejo, M. (1 December 1997). Evaluation of the rotation matrices in the basisof real spherical harmonics. Journal of Molecular Structure: THEOCHEM 419 (13): 1927. doi:10.1016/S0166-1280(97)00185-1.

General references

See section 3 in Mathar, R. J. (2009). Zernike basis to cartesian transformations. Serbian Astronomical Jour-nal 179 (179): 107120. arXiv:0809.2368. Bibcode:2009SerAj.179..107M. doi:10.2298/SAJ0979107M.(see section 3.3)

For complex spherical harmonics, see also SphericalHarmonicY[l,m,theta,phi] at Wolfram Alpha, especiallyfor specic values of l and m.

Chapter 6

Toronto function

In mathematics, the Toronto function T(m,n,r) is a modication of the conuent hypergeometric function denedby Heatley (1943) as

T (m;n; r) = r2nm+1er2 ( 12m+

12 )

(n+ 1)1F1(

12m+

12 ;n+ 1; r

2):

6.1 References Heatley, A. H. (1943), A short table of the Toronto function, Trans. Roy. Soc. Canada Sect. III. 37: 1329,MR 0010055

36

Chapter 7

Trigonometric integral

Sine integral and cosine integral

x0 5 10 15 20 25

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Si(x)Ci(x)

Si(x) (blue) and Ci(x) (green) plotted on the same plot.

In mathematics, the trigonometric integrals are a family of integrals involving trigonometric functions. A numberof the basic trigonometric integrals are discussed at the list of integrals of trigonometric functions.

7.1 Sine integralThe dierent sine integral denitions are

Si(x) =Z x0

sin tt

dt

37

38 CHAPTER 7. TRIGONOMETRIC INTEGRAL

Plot of Si(x) for 0 x 8 .

si(x) = Z 1x

sin tt

dt :

By denition, Si(x) is the antiderivative of sin x / x which is zero for x = 0; and si(x) is the antiderivative of sin x / xwhich is zero for x = . Their dierence is given by the Dirichlet integral,

Si(x) si(x) =Z 10

sin tt

dt =

2:

Note that sin x / x is the sinc function, and also the zeroth spherical Bessel function.In signal processing, the oscillations of the sine integral cause overshoot and ringing artifacts when using the sinclter, and frequency domain ringing if using a truncated sinc lter as a low-pass lter.Related is the Gibbs phenomenon: if the sine integral is considered as the convolution of the sinc function with theheaviside step function, this corresponds to truncating the Fourier series, which is the cause of the Gibbs phenomenon.

7.2 Cosine integralThe dierent cosine integral denitions are

Ci(x) = Z 1x

cos tt

dt = + lnx+Z x0

cos t 1t

dt

Cin(x) =Z x0

1 cos tt

dt ;

7.3. SINHC FUNCTION INTEGRAL 39

Plot of Ci(x) for 0 < x 8.

where is the EulerMascheroni constant. Some texts use ci instead of Ci.Ci(x) is the antiderivative of cos x / x (which vanishes at x = ). The two denitions are related by

Cin(x) = + lnx Ci(x) :

7.3 Sinhc function integralThe hyperbolic sine integral is dened as

Shi(x) =Z x0

sinh(t)t

dt:

7.4 Hyperbolic cosine integralThe hyperbolic cosine integral is

Chi(x) = + lnx+Z x0

cosh t 1t

dt = chi(x)

where is the Euler-Mascheroni constant.It has the series expansion Chi(x) = + ln(x) + 14x2 + 196x4 + 14320x6 + 1322560x8 + 136288000x10 +O(x12) .


7.5 Auxiliary functionsTrigonometric intervals can be understood in terms of the so-called auxiliary functions

f(x) Z 10

sin(t)t+ x

dt =

Z 10

ext

t2 + 1dt = Ci(x) sin(x) +

h2 Si(x)

icos(x)

g(x) Z 10

cos(t)t+ x

dt =

Z 10

text

t2 + 1dt = Ci(x) cos(x) +

h2 Si(x)

isin(x)

Using these functions, the trigonometric integrals may be re-expressed as (cf Abramowitz & Stegun, p. 232)

Si(x) = 2 f(x) cos(x) g(x) sin(x)Ci(x) = f(x) sin(x) g(x) cos(x):

7.6 Nielsens spiral

Nielsens spiral.

The spiral formed by parametric plot of si , ci is known as Nielsens spiral. It is also referred to as the Euler spiral,the Cornu spiral, a clothoid, or as a linear-curvature polynomial spiral.The spiral is also closely related to the Fresnel integrals. This spiral has applications in vision processing, road andtrack construction and other areas.

7.7 ExpansionVarious expansions can be used for evaluation of Trigonometric integrals, depending on the range of the argument.

7.8. RELATION WITH THE EXPONENTIAL INTEGRAL OF IMAGINARY ARGUMENT 41

7.7.1 Asymptotic series (for large argument)

Si(x) = 2 cosx

x

1 2!

x2+

4!

x4 6!

x6 sinx

x

1

x 3!

x3+

5!

x5 7!

x7

Ci(x) = sinxx

1 2!

x2+

4!

x4 6!

x6 cosx

x

1

x 3!

x3+

5!

x5 7!

x7

:

These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at (x) 1.

7.7.2 Convergent series

Si(x) =1Xn=0

(1)nx2n+1(2n+ 1)(2n+ 1)!

= x x3

3! 3 +x5

5! 5 x7

7! 7

Ci(x) = + lnx+1Xn=1

(1)nx2n2n(2n)!

= + lnx x2

2! 2 +x4

4! 4

These series are convergent at any complex x, although for |x | 1 the series will converge slowly initially, requiringmany terms for high precisions.

7.8 Relation with the exponential integral of imaginary argumentThe function

E1(z) =Z 11

exp(zt)t

dt ( 0) :

As each respective function is analytic except for the cut at negative values of the argument, the area of validity of therelation should be extended to (x) > 0. (Outside this range, additional terms which are integer factors of appearin the expression.)Cases of imaginary argument of the generalized integro-exponential function are

Z 11

cos(ax) lnxx

dx = 2

24+

2+ ln a

+

ln2 a2

+Xn1

(a2)n(2n)!(2n)2

;

which is the real part of

Z 11

eiaxlnxx

dx = 2

24+

2+ ln a

+

ln2 a2

2i( + ln a) +

Xn1

(ia)n

n!n2:

Similarly

Z 11

eiaxlnxx2

dx = 1 + ia[2

24+

2+ ln a 1

+

ln2 a2

ln a+ 1 i2( + ln a 1)] +

Xn1

(ia)n+1

(n+ 1)!n2:


7.9 Ecient evaluation

Pad approximants of the convergent Taylor series provide an ecient way to evaluate the functions for small argu-ments. The following formulae are accurate to better than 1016 for 0 x 4,

Si(x) = x

0BBBBBB@1 4:54393409816329991 102 x2 + 1:15457225751016682 103 x4 1:41018536821330254 105 x6+ 9:43280809438713025 108 x8 3:53201978997168357 1010 x10 + 7:08240282274875911 1013 x12 6:05338212010422477 1016 x14

1 + 1:01162145739225565 102 x2 + 4:99175116169755106 105 x4 + 1:55654986308745614 107 x6+ 3:28067571055789734 1010 x8 + 4:5049097575386581 1013 x10 + 3:21107051193712168 1016 x12

1CCCCCCACi(x) = + ln(x)+

x2

0BBBBBB@0:25 + 7:51851524438898291 103 x2 1:27528342240267686 104 x4 + 1:05297363846239184 106 x6 4:68889508144848019 109 x8 + 1:06480802891189243 1011 x10 9:93728488857585407 1015 x12

1 + 1:1592605689110735 102 x2 + 6:72126800814254432 105 x4 + 2:55533277086129636 107 x6+ 6:97071295760958946 1010 x8 + 1:38536352772778619 1012 x10 + 1:89106054713059759 1015 x12+ 1:39759616731376855 1018 x14

1CCCCCCAFor x > 4, instead, one can use the above auxiliary functions f(x) and g(x). Chebyshev-Pad expansions of 1py f

1py

and 1y g

1py

in the interval (0, 1/42] yield the following approximants, good to better than 1016 for x 4 :

f(x) =1

x

0BBBBBBBBBB@

1 + 7:44437068161936700618 102 x2 + 1:96396372895146869801 105 x4 + 2:37750310125431834034 107 x6+ 1:43073403821274636888 109 x8 + 4:33736238870432522765 1010 x10 + 6:40533830574022022911 1011 x12+ 4:20968180571076940208 1012 x14 + 1:00795182980368574617 1013 x16 + 4:94816688199951963482 1012 x18 4:94701168645415959931 1011 x20

1 + 7:46437068161927678031 102 x2 + 1:97865247031583951450 105 x4 + 2:41535670165126845144 107 x6+ 1:47478952192985464958 109 x8 + 4:58595115847765779830 1010 x10 + 7:08501308149515401563 1011 x12+ 5:06084464593475076774 1012 x14 + 1:43468549171581016479 1013 x16 + 1:11535493509914254097 1013 x18

1CCCCCCCCCCA

g(x) =1

x2

0BBBBBBBBBB@

1 + 8:1359520115168615 102 x2 + 2:35239181626478200 105 x4 + 3:12557570795778731 107 x6+ 2:06297595146763354 109 x8 + 6:83052205423625007 1010 x10 + 1:09049528450362786 1012 x12+ 7:57664583257834349 1012 x14 + 1:81004487464664575 1013 x16 + 6:43291613143049485 1012 x18 1:36517137670871689 1012 x20

1 + 8:19595201151451564 102 x2 + 2:40036752835578777 105 x4 + 3:26026661647090822 107 x6+ 2:23355543278099360 109 x8 + 7:87465017341829930 1010 x10 + 1:39866710696414565 1012 x12+ 1:17164723371736605 1013 x14 + 4:01839087307656620 1013 x16 + 3:99653257887490811 1013 x18

1CCCCCCCCCCAHere are text versions of the above suitable for copying into computer code (using x2 = x*x and y = 1/(x*x) whereappropriate):Si = x*(1. + x2*(4.54393409816329991e-2 + x2*(1.15457225751016682e-3 + x2*(1.41018536821330254e-5+ x2*(9.43280809438713025e-8 + x2*(3.53201978997168357e-10 + x2*(7.08240282274875911e-13 + x2*(6.05338212010422477e-16)))))))) / (1. + x2*(1.01162145739225565e-2 + x2*(4.99175116169755106e-5 + x2*(1.55654986308745614e-7+ x2*(3.28067571055789734e-10 + x2*(4.5049097575386581e-13 + x2*(3.21107051193712168e-16))))))) Ci =0.577215664901532861 + ln(x) + x2*(0.25 + x2*(7.51851524438898291e-3 + x2*(1.27528342240267686e-4+ x2*(1.05297363846239184e-6 + x2*(4.68889508144848019e-9 + x2*(1.06480802891189243e-11 + x2*(9.93728488857585407e-15))))))) / (1. + x2*(1.1592605689110735e-2 + x2*(6.72126800814254432e-5 + x2*(2.55533277086129636e-7 +x2*(6.97071295760958946e-10 + x2*(1.38536352772778619e-12 + x2*(1.89106054713059759e-15 + x2*(1.39759616731376855e-18)))))))) f = (1. + y*(7.44437068161936700618e2 + y*(1.96396372895146869801e5 + y*(2.37750310125431834034e7+ y*(1.43073403821274636888e9 + y*(4.33736238870432522765e10 + y*(6.40533830574022022911e11 + y*(4.20968180571076940208e12+ y*(1.00795182980368574617e13 + y*(4.94816688199951963482e12 + y*(4.94701168645415959931e11)))))))))))/ (x*(1. + y*(7.46437068161927678031e2 + y*(1.97865247031583951450e5 + y*(2.41535670165126845144e7+ y*(1.47478952192985464958e9 + y*(4.58595115847765779830e10 + y*(7.08501308149515401563e11 + y*(5.06084464593475076774e12+ y*(1.43468549171581016479e13 + y*(1.11535493509914254097e13))))))))))) g = y*(1. + y*(8.1359520115168615e2+ y*(2.35239181626478200e5 + y*(3.12557570795778731e7 + y*(2.06297595146763354e9 + y*(6.83052205423625007e10+ y*(1.09049528450362786e12 + y*(7.57664583257834349e12 + y*(1.81004487464664575e13 + y*(6.43291613143049485e12

7.10. SEE ALSO 43

+ y*(1.36517137670871689e12))))))))))) / (1. + y*(8.19595201151451564e2 + y*(2.40036752835578777e5 +y*(3.26026661647090822e7 + y*(2.23355543278099360e9 + y*(7.87465017341829930e10 + y*(1.39866710696414565e12+ y*(1.17164723371736605e13 + y*(4.01839087307656620e13 + y*(3.99653257887490811e13))))))))))

7.10 See also Exponential integral Logarithmic integral

7.10.1 Signal processing Gibbs phenomenon Ringing artifacts

7.11 References Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 5, Handbook of Mathematical Functionswith Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 231, ISBN 978-0486612720, MR0167642.

Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), Section 6.8.2. Cosine and Sine Integrals,Numerical Recipes: The Art of Scientic Computing (3rd ed.), New York: Cambridge University Press, ISBN978-0-521-88068-8

Temme, N. M. (2010), Exponential, Logarithmic, Sine, and Cosine Integrals, in Olver, Frank W. J.; Lozier,Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, CambridgeUniversity Press, ISBN 978-0521192255, MR 2723248

Mathar, R. J. (2009). Numerical evaluation of the oscillatory integral over exp(ix)x1/x between 1 and ".arXiv:0912.3844., Appendix B.

Sine Integral Taylor series proof from Dan Sloughters Dierence Equations to Dierential Equations.

7.12 External links http://mathworld.wolfram.com/SineIntegral.html Hazewinkel, Michiel, ed. (2001), Integral sine, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Hazewinkel, Michiel, ed. (2001), Integral cosine, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4


7.13 Text and image sources, contributors, and licenses7.13.1 Text

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StieltjesWigert polynomials Source: https://en.wikipedia.org/wiki/Stieltjes%E2%80%93Wigert_polynomials?oldid=646860010Con-tributors: Michael Hardy, R.e.b., Sodin, Headbomb, Leyo, Yobot, Specfunfan and Omnipaedista

Table of spherical harmonics Source: https://en.wikipedia.org/wiki/Table_of_spherical_harmonics?oldid=638091596 Contributors:Cyp, Jitse Niesen, Almit39, Linas, Kkmurray, SmackBot, Maksim-e~enwiki, InverseHypercube, Bduke, Chlewbot, William Ackerman,Xxanthippe, Headbomb, JAnDbot, Baccyak4H, Lunokhod, Thurth, PolarBot, Romzromz, BOTarate, Addbot, WuBot, Luckas-bot, Cita-tion bot, Obersachsebot, SassoBot, Erik9bot, Sawomir Biay, Wagnerif, Oakycoppice, Superlaser1, ZroBot, R. J. Mathar, MerlIwBot,Bibcode Bot, ThaeliosActual, Loudandras, Susilehtola, Monkbot, Ying.l.xiong and Anonymous: 30

Toronto function Source: https://en.wikipedia.org/wiki/Toronto_function?oldid=627090821Contributors: Michael Hardy, R.e.b., Yobot,Trappist the monk and Adairre 1010

Trigonometric integral Source: https://en.wikipedia.org/wiki/Trigonometric_integral?oldid=669241952 Contributors: Damian Yerrick,Lir, Michael Hardy, Notheruser, Stan Lioubomoudrov, MathMartin, Giftlite, Monedula, Alberto da Calvairate~enwiki, MartinBiely, Nor-nagon~enwiki, PAR, Linas, Gisling, R.e.b., Mathbot, YurikBot, NTBot~enwiki, Reyk, SmackBot, Nbarth, Ascentury, Dream out loud,Sammy1339, Domitori, Curgny, ShelfSkewed, Eric Le Bigot, Johner, Kupirijo, Thijs!bot, AmitAronovitch, DmitTrix, Second Quantiza-tion, LachlanA, .anacondabot, Robert Illes, Policron, Fylwind, Cuzkatzimhut, Yugsdrawkcabeht, Flyer22, Rmjarvis, Ideal gas equation,Tom363, Muro Bot, Joctee, Torchame, Addbot, Dr. Universe, PV=nRT, Legobot, Yobot, AnomieBOT, FrescoBot, AManWithNoPlan,Versatranitsonlywaytoy, MerlIwBot, ServiceAT, Randomguess and Anonymous: 31

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