BVRIT HYDERABAD College of Engineering for Women

Department of Mathematics

Course Hand Out

Subject Name : MATHEMATICS -IV (MA301BS)

Year, Semester, Regulation : II year, I semester, R16

Faculty Name : Mr.A.Sampath Kumar and Ms.B.Sreekala

UNIT – I Functions of a complex variable: Introduction, Continuity, Differentiability,

Analyticity, properties, Cauchy, Riemann equations in Cartesian and polar

coordinates. Harmonic and conjugate harmonic functions-Milne-Thompson

method

UNIT - II Complex integration: Line integral, Cauchy’s integral theorem, Cauchy’s

integral formula, and Generalized Cauchy’s integral formula, Power series:

Taylor’s series- Laurent series, Singular points, isolated singular points, pole of

order m – essential singularity, Residue, Cauchy Residue theorem (Without

proof).

UNIT – III Evaluation of Integrals: Types of real integrals:

a) Improper real integrals

dxxf )( b)

2

)sin,(cos

c

c

df

Bilinear transformation- fixed point- cross ratio- properties- invariance of circles.

UNIT – IV Fourier series and Transforms: Introduction, Periodic functions, Fourier series

of periodic function, Dirichlet’s conditions, Even and odd functions, Change of

interval, Half range sine and cosine series. Fourier integral theorem (without proof), Fourier sine and cosine integrals, sine

and cosine, transforms, properties, inverse transforms, Finite Fourier transforms.

UNIT – V Applications of PDE: Classification of second order partial differential equations,

method of separation of variables, Solution of one dimensional wave and heat

equations.

TEXT BOOKS: 1. A first course in complex analysis with applications by Dennis G. Zill and

Patrick Shanahan, Johns and Bartlett Publishers. 2. Higher Engineering Mathematics by Dr. B. S. Grewal, Khanna Publishers. 3. Advanced engineering Mathematics with MATLAB by Dean G. Duffy

REFERENCES: 1. Fundamentals of Complex Analysis by Saff, E. B. and A. D. Snider, Pearson. 2. Advanced Engineering Mathematics by Louis C. Barrett, McGraw Hill.

SOME TRIGNOMETRIC FORMULAE:

0 30 45 60 90

sin 0 1/2 1/2 3/2 1

cos 1 3/2 1/2 1/2 0

tan 0 1/3 1 3

2 2

2 2

2 2

sin cos 1

1 tan sec

1 cot sec

x x

x x

x co x

2

2

1 cos 2sin

2

1 cos 2cos

2

xx

xx

3 1sin 3sin sin3

4x x x 3 1

cos 3cos cos34

x x x

sin( ) sin cos cos sin

sin( ) sin cos cos sin

cos( ) cos cos sin sin

cos( ) cos cos sin sin

A B A B A B

A B A B A B

A B A B A B

A B A B A B

2sin cos sin( ) sinA B A B A B

2cos sin sin( ) sinA B A B A B 2cos cos cos cosA B A B A B

2sin sin cos cosA B A B A B

cosh2

sinh2

ax ax

ax ax

e eax

e eax

DIFFERENTIATION FORMULAE:

1

( ) 0

( )

( ) log

( )

n n

x x

x x

dK

dx

dx nx

dx

da a a

dx

de e

dx

2

2

1

2

1

2

1

2

1

2

1

2

(sin ) cos

(cos ) sin

(tan ) sec

(cot ) cos

(sec ) sec tan

(cos ) cos cot

1(sin )

1

1(cos )

1

1(sec )

1

1(cos )

1

1(tan )

1

dx x

dx

dx x

dx

dx x

dx

dx ec x

dx

dx x x

dx

decx ecx x

dx

dx

dx x

dx

dx x

dx

dx x x

dec x

dx x x

dx

dx x

d

dx

1

2

1(cot )

1x

x

2

2

(sinh ) cosh

(cosh ) sinh

(tanh ) sec

(coth ) cos

dx x

dx

dx x

dx

dx h x

dx

dx ech x

dx

2

( )

( )

( )

d dKu K u

dx dx

d d du v u v

dx dx dx

d d duv u v v u

dx dx dx

d dv u u v

d u dx dx

dx v v

[ ( )] [ ( )] ( )d

f g x f g x g xdx

INTEGRATION FORMULAE:

k dx k x c

1

, 11

nn x

x dx c nn

1

logdx x cx

log logxdx x x x c

log

xx a

a dx ca

x xe dx e c

sin cosxdx x c

cos sinxdx x c

2sec tanx dx x c

2cos dx = cotx +cec x

sec tan secx xdx x c

cos cot cosecx xdx ecx c

tan log cosxdx x c or

log sec x c

cot log sin xdx x c

sec log sec tan or log tan4 2

xxdx x x c c

cos log cos cot log tan2

xecxdx ecx x c or c

2

2

  sinh cosh  

  cosh sinh    

  tanh logcosh     

  coth logsinh   

  sech tanh  

  cos h coth  

xdx x c

xdx x c

xdx x c

xdx x c

xdx x c

ec xdx x c

1 1

2

1sin + c cos + c

1dx x or x

x

1 1

2

1tan cot

1dx x c or x c

x

1 1

2

1sec cos

1dx x c or ec x c

x x

1 1

2 2

1sin cos

x xdx c or c

a aa x

1 1

2 2

1 1 1tan cot

x xdx c or c

a x a a a a

1 1

2 2

1 1 1sec sec

x xdx c or co c

a a a ax x a

2 2

2 2

1logdx x x a c

x a

2 2

2 2

1logdx x x a c

x a

2 2

1 1log

2

x adx c

x a a x a

2 2

1 1log

2

a xdx c

a x a a x

2 2 22 2 1sin

2 2

x x a a xa x dx c

a

2 2 22 2 2 2log

2 2

x x a ax a dx x x a c

2 2 22 2 2 2log

2 2

x x a ax a dx x x a c

2 2

2 2

sin sin cos

cos cos sin

axax

axax

ee bx dx a bx b bx

a b

ee bx dx a bx b bx

a b

INTEGRATION BY PARTS:

Integration by parts is used in integrating product of functions of the type .f x g x as follows:

st nd st nd

st nd

I function II function dx I function II function dx

dI function II function dx dx

dx

Where the Ist and II

nd functions are decided in the order of ILATE;

I: Inverse trigonometric function

L: Logarithmic function

A: Algebraic functions

T: Trigonometric functions

E: Exponential Functions

I 1 2 1 2f x f x dx f x dx f x dx

II .k f x dx k f x dx

III

1

, 11

n

n ax bax b dx c n

a n

log1 ax b

dx cax b a

ax b

ax b ee dx c

a

cos

sinax b

ax b dx ca

etc

IV

1( )

.1

nn f x

f x f x dxn

+c

V

log

f xdx f x c

f x

VI

2

f xdx f x c

f x

VII x xe f x f x dx e f x c

VIII ( )

f x f xe f x dx e c

Power series:

i) 3 5

sinx  ......3! 5!

x xx

ii) 2 4

cosx 1 ......2! 4!

x x …………..

iii) 2 3

log 1 x ......2! 3!

x xx

iv) 2 3

xe 1 .........2! 3!

x xx ……….

v) 3 5

sinhx .........3! 5!

x xx

vi) 2 4

coshx 1 ......2! 4!

x x

UNIT –I

(Functions of Complex Variable)

Defn: A number of the form x+iy , where x and y are real numbers and i= )1( is called a

complex number.x is jcalled the real part of x+iy and y is called the imaginary part

written R(x+iy), I(x+iy) respectively.

Properties:

1) If x+iy=u+iv then x-iy=u-iv

2) Two complex numbers x+iy and u+iv are said to be equal where R(x+iy)=R(u+iv) is x=u,

I(x+iy)=I(u+iv) i.e y=v

3) Sum, difference, product and quotient of any complex numbers is itself a csomplex

number.

4) Every complex number x+iy can always be expressed in the form )sin(cos ir

Defn: The number 22 yxr is called the modulus of x+iy and is written as

mod(x+iy) or (x+iy) the angle is called the amplitude of argument of x+iy and is written

as amp(x+iy) or arg(x+iy).

Evidently, the amplitude has an infinite number of values. The value of which lie

between – and is called the principal value of the Amplitude.

If the conjugate of Z = x+iy be Z then R(Z) = )(21 ZZ and I(Z) = )(

21 ZZi

|Z| = )()( 22 ZIZR = | Z |

|Z|2= Z Z

2121 ZZZZ

2121 ZZZZ

2

1

2

1

Z

Z

Z

Z

where 2Z 0

The point whose Cartesian coordinates are (x,y) uniquely represents the complex

number z = x+iy on the complex plane Z. The diagram in which the representation is

carried out is called the argand’s diagram.

If Z1,Z2 are two complex numbers then

1. 2121 ZZZZ

2. 2121 ZZZZ

In general |Z1+Z2+……+Zn| |Z1| + |Z2| +……+ |Zn|

3. amp (Z1Z2) = amp (Z1) + amp (Z2)

4. |Z1/Z2|=|Z1| / |Z2|

5. amp (Z1 / Z2) = amp (Z1)- amp(Z2)

Demoivre’s theorem: If n be I) an integer positive or negative then (Cos +isin)n =

cos n +isinn

If x and y are real variables then z = x+iy is called complex variable. If corresponding to each

value of the complex variable z(= x+iy) in a region R there corresponds one or more values of

another complex variable w (= u+iv) then w is called a function of complex variable z and is

denoted by w =f(z) = u+iv where u,v are real and imaginary parts of w and the function of real

variables w =f(z) =u(x,y)+iv(x,y)

If to each value of z there corresponds one and only one value of w then w is called a

single valued function of z.

If to each value of z there corresponds more than one value of w then w is called multi

valued function of z

To represent w =f(z) graphically, we take two argand diagrams one to represent the point z and

the other to represent the point w

The distance between the point z and ‘a’ is denoted by |z-a|

A circle of radius‘d’ with center at ‘a’ is denoted by |z-a| = d.

The inequality |z-a|< d denoted by every point inside the circle C: |z-a|<d i.e., it represents the

interior of the circle excluding its circumference. The interior of the circle including its

circumference is denoted by |z-a| d.

The Neighborhood of a point ‘a’ is represented by the inequality |z-a|<d.

|z-a| >d represents the exterior of the circle with center at ‘a’ and radius ‘d’.

The region between two concentric circles of radii d1 and d2 ( d1 > d2) can be represented by

d1 <|z-a| <d2

The equation |z| =1 represents a unit circle about origin.

If there exists a circle with center at origin enclosing all points of a region R then R is said to be

bounded.

If a region is defined to include all the points on its various boundary curves, it is said to be

closed.

If R contains none of its boundary points, it is said to be open.

A set of points in the complex plane S is called open if every point of S has a ngd. All the points

of which belong to S.

A set of points in the complex plane S is called closed if the points which do not belong to S form

an open set S.

Limit of f(z): A function w= f(z) tends to the limit ‘’ as z approaches a point z0 along any

path, if to ‘ each positive arbitrary number , however small there corresponds a positive number ,

such that |f(z)- | < whenever 0<|z- z0|<

i.e., - < f(z) < + whenever z0- <z< z0+ , z z0

We write

lim0zz

In real variables x x0 implies x approaches x0 along the line either from left or right.

In complex variables z z0 implies z approaches z0 along the path (straight or curved) since a

complex plane can be joined by infinite number of curves.

Continuity of f(z): A single valued function f(z) is said to be continuous at a point z = z0 if

lim

0zz

f(z0)

A function f(z) is said to be continuous in a region R in the Z-plane if it is continuous at every

point of the region.

If w= f(z)= U(x,y)+iv(x,y) is continuous at z= z0 then u(x,y) and v(x,y) are also continuous at z=

z0 i.e., at x= x0 and y=y0.

Conversely, if u(x,y) and v(x,y) are continuous at (x0,y0) then f(z) will be continuous at z=z0

Sum, difference and product of two continuous functions is continuous .Quotient function of two

continuous if exists then it is also continuous .If f(z) is continuous |f(z)| is also continuous.

Differentiablity:

A single valued function f(z) is differentiable at the point z=z0 is denoted by f ‘(z) or dz

dw and is

defined by the equation f 1(zo)=

z

zfzzf

z

)()(lim

0

provided the limit exists.

Analytic function: A single valued function f(z) is said to be analytic at a point z0 if it has a unique

derivative at z0 and at every point in the neighborhood of z0

Cauchy-Riemann Equations:

Cartesian form: The necessary and sufficient condition for for function w=f(z) to be analytic in a

Region R are a) The four first order derivatives y

v

x

v

y

u

x

u

,,, exists and are continuous in R.

b) y

v

x

u

and

x

v

y

u

The conditions given in b) are called cauchy-Riemann equation or C.R. Equations.

Polar form: Let (r,) be the polar co-ordinates of the point whose Cartesian co-ordinates are (x,y)

with x = r cos y = r sin.The C.R. Equations are

v

rr

u 1 and

r

vr

u

Harmonic Function: Any function (x,y) which possess continuous partial derivatives of the first

and second orders and satisfy Laplace equations 02

2

2

2

yx

is called Harmonic function.

Conjugate Harmonic function: If a function u(x,y) is Harmonic in the domain and if we find

another Harmonic function v(x,y) such that they satisfy the cauchy- Riemann equations and Laplace

equations then we say v(x,y) is harmonic conjugate of u(x,y).

Properties of Analytic Functions:

An analytic function with constant real part is constant.

An analytic function with constant imaginary part is constant.

An analytic function with constant modulus is constant.

The real and imaginary parts of an analytic functions are harmonic

Every analytic function f(z) = u+iv defines two families of curves u(x,y) = c1 and

v(x,y) =c2 forms an orthogonal system.

An analytic function can be easily constructed by using Milne –Thomson method.

Complex potential function: The analytic function w = (x,y)+i(x,y) is called complex potential

function. Its real part (x,y) represents the velocity potential function and its imaginary part (x,y)

represents the stream function.

Both , satisfy Laplace equation. Given any one of them we find the other.

Objective Questions:

1. The value of m so that 222 myxx may be harmonic is ( )

A) 0 B) 1 C) 2 D) 3

2. The necessary condition for , ,f z u x y iv x y to be analytic is ( )

A)u v u v

andx y y x

B)

u v u vand

x y y x

C) u v u v

andx y y x

D)

u v u vand

x x y y

3. Which of the following is an entire function ( )

A) 1z

B) ze C) tan z D) log z

4. The analytic function for which the real part cosxe y is ( )

A) 1

ze B) ze C)

ze D)

1ze

5. If u iv is analytic, then u iv is ( )

A) analytic B) not analytic C) analytic at origin D) analytic everywhere except origin

6. The function ,x y that satisfies Laplace equation is known as____________________

7. An analytic function with constant absolute value (or modulus) is ____________________

8. Polar form of Cauchy-Riemann equations are____________________________

9. The singular point of the function 2

3

2 1

8

z zf z

z

is ____________________________

10. A point at which f(z) fails to be analytic is ____________________

UNIT-II

(COMPLEX INTEGRATION)

Complex Line integral: Let f(z) be a function which is continous at all points on the curve C

whose end points are A,B

Dividing the curve C into n parts by the points z0 (= A),z1,z2,…..zn (=B). Let f(z) be defined at

all these points.Let zr be a point on the arc joining zr-1 to zr. Let zr-zr-1= zr. Define the sum Sn =

n

r

rr zf

1

)( the limit of the sum Sn as n tents to infinity and zr tends to zero if exists is

denoted by C

b

a

dzzfordzzf )()( . This is called the line integral of f(z) along the curve C.

Closed Curve: If the points z0 and zn coincide then curve C is closed curve.

The integral of closed curve is called the contour integral and is denoted by C dzzf )(

Relation between real and complex line integrals: If Z= x+iy so that dz=dx+idy and f(z)=

u(x,y) +iv(x,y) then the complex line integral C

dzzf )( can be expressed as sum or difference

of two line integrals of real functions as under

C

dzzf )( =

C

vdyudx )( + i

C

udyvdx =

C

idydxivu ))((

If f(z) =1 then we have C

dz || = C

ds = where is the length of the path of integration.

If C is a closed curve then C

dz =0

If C is a circle of radius r and center z0 and if n is an integer then

Cn

nzz

dz0,0

)( 10

= 2 i, n=0

Simply Connected Region: A region is said to be simply connected if any simple closed curve

lying in R can be shrunk to a point without leaving R

Multiply connected region: A region that is not simply connected is called multiply connected

region.

Cauchy’s Integral Theorem: If f(z) is analytic function and f1(z)is continuous at each point with

in or on a closed curve C then C

dzzf )( =0

Extension of Cauchy’s Integral theorem: If C1 and C2 are two simple closed curves and if C2

lies entirely within the closed region between C1 and C2 then 1

)(

C

dzzf = 2

)(

C

dzzf both the

integrals are taken in the same direction

If there are finite number of contours C1,C2……Cn with in C and f(z) is analytic in the region

with in the region between C1,C2……Cn then we have

C

dzzf )( = 1

)(

C

dzzf + 2

)(

C

dzzf +………+ nC

dzzf )( provided all the integrals are taken

in same direction.

Cauchy’s Integral formula: If f(z) is an analytic function inside and on a simple closed curve C

and z0 is any point within C then f(z0)= dz

zz

zf

i )(

)(

2

1

0

Derivative of an Analytic function: f(z0) = dz

zz

zf

i

n

n 10 )(

)(

2

!

Infinite series- Taylor’s and Laurent’s series:

Taylor’s series : If a function f(z) is Analytic inside a circle ‘c’ whose center is ‘a’ then for all z

inside c f(z)=

)(!

)()(

!2

)()()()(

2

afn

azaf

azafazaf n

n

1. Put z=a+h (or) h=z-a

f(a+h)= )(!

)(!2

)()(2

afn

haf

hafhaf n

n

2. Put a=0 f(z)= )0(!

)0(!2

)0()0(2

nn

fn

zf

zfzf is called Maclaurin’s

series.

Laurent’s series:

If f(z) is analytic inside and on the boundary of the ring stated region R bounded by two concentric

circles c1 and c2 of radii r1 and r2 ( r1 > r2) respectively having center at ‘a’ then for all z in R

F(z)=

22

11

2210 )()()()( azaazaazaazaa

Where an= 1

1)(

)(

2

1

cn

dwaw

wf

i , n=0,1,2,-------

and a –1= 2

1)(

)(

2

1

cn

dwaw

wf

i,n=1,2,3,-----

Zeros and Singularities:

Zeros of an Analytic function: A zero of an Analytic function f(z) is that value of z for which

f(z)=0

Singularities of an Analytic function : A singularity of a function is that point at which the

function f(z) ceases to be analytic.

Isolated Singularity : If z=a is a singularity of f(z) and if f(z) is analytic at each point in its

neighbourhood then z=a is called an isolated singularity.

Removable singularity :f(z)=

0

)(

n

nn aza The singularity can be removed by defining the

function f(z) at z=a in such a way that it becomes analytic at z=a.

Poles: If all the negative powers of (z-a) in

f(z)=

22

11

2210 )()()()( azaazaazaazaa

after the nth

we missing then the singularity at z=a is called a pole of order n.

A pole of first order is called a Simple pole.

Essential Singularity: If the number of negative powers of (z-a) in

f(z)=

22

11

2210 )()()()( azaazaazaazaa

is in finite then z=a is called an essential singularity in this case )(lim zf

az

does not exist .

Residues: The coefficient of (z-a) in the expansion of f(z) around an isolated singularity is called the

residue of f(z) at that point and is written as Re ( )z a

s f z

Evaluation of Residues:

1. If f(z) has a simple pole at z=a then Re ( )z a

s f z

= )()(lim zfaz

az

2. Suppose f(z) =)(

)(

z

z

where )()()( zFazz where F(a) 0 then Re ( )

z as f z

=

)(

)(

a

a

3. Let z=a be a pole of f(z) of order m then

Re ( )z a

s f z

= )()()1(

1

1

1

lim zfazdz

d

m

m

m

m

az

Cauchy’s Residue theorem:

If f(z) is analytic in a closed curve C except at a finite number of singular points within C then

C

dzzf )( = 2i( Sum of the residues at the singular point within C)

Objective Questions:

1. The value of

23

z

C

edz

z over the circle 1 1z is ( )

A) 23

i B) 0 C) 43

i D) 43

i

2. The Residue of

23

zze

z is ( )

A) 24e B) 22e C)

34e D) 25e

3. Expand

1

2z z when 2z ( )

A)2 31

1 .......2 2 4 8

z z z

z

B)

2 311 .......

2 2 4 8

z z z

z

C) 2 31

1 .......2 2 4 8

z z z

z

D)

2 31 2 3 41 .......

2 2 4 8

z z z

z

4. The poles of 1 2

zf z

z z

are ( )

A) 1& 2z z B) 1& 2z z C) 1& 2z z D) 1& 2z z

5. Expand log 1f z z in a Taylor’s series about 0z

____________________________

6. The value of

2

3

z

C

edz

z z over the circle 2z is____________________________

7. If the power series 0

n

n

n

a z

converges for z R and diverges for z R then z R is

called____________________________

8.The value of C

z dz where C is the contour consisting of the straight line from z i to z i

____________________________

9. The series 2 31 ........z z z is equal to _________________

10.

2

2

1,

4c

dzz

where : 1C z is _______________________

UNIT-III

(Evaluation of Integrals)

Evaluation of some of the definite integrals:

Many of the definite integrals can be evaluated by using cauchy’s

residue theorem. It may be observed that a definite integral that can be evaluated by using

Cauchy’s residue theorem may also be evaluated by other methods namely:

I) Integration around the unit circle

II) Integration of the type

dxxf )(

III) Indenting contours having poles on real axis

IV) Using Jordan’s Lemma

Mapping Or Transformation: A curve C in the z-plane is mapped into the corresponding curve

C1 in the w-plane by the function w =f(z) which defines a mapping or transformation of the z-

plane into the w-plane.

Conformal and Isogonal Transformations: Let C & C1 be the two curves in the z-plane which

intersect at the point P. Let C1

& C11 be the images of C & C1 in the w –plane which intersect

at the point P1. If the transformation is such that the angle between C & C1 at P is equal in

magnitude and sense to the angle between C1 & C1

1 at P

1 then the transformation is called

Conformal.The mapping is said to be isogonal if it preserves the magnitudes of the angles but not

sense.

The points where f1(z) = 0 are called the critical points of the transformation.

Elementary Transformations :

Translation: w = z + c where c is a complex constant. This transformation merely

translates every point z through the constant vector c.

Rotation & Magnification : w = cz where c is a complex constant. Let c = ei

,

z = rei

, w= Rei then transformations becomes

Rei

= rRrei )( and hence the transformation maps point

p(r, ) in the z-plane into the point P1(r , ) in the w-plane.

Inversion and Reflection: Inversion is represented by w = z

1 and it maps the interior of the

unit circle |z| = 1 onto the exterior of the unit circle |w| = 1and the exterior of |z| = 1 onto the

interior |w| =1.

It maps a circle onto the circle and this transformation is an inversion of z w.r.t the unit circle |z|

= 1 followed by reflection of the inverse into the real axis .

Bilinear Transformation or Mobius Transformation: A Transformation of the form w =

dcz

baz

where a ,b, c, d are complex constants such that ad-bc 0 is called a Bilinear

Transformation.

Cross Ratio: The cross ratio of four points z1, z2, z3, z4 taken in this order defined as

))((

))((

1432

4321

zzzz

zzzz

and is denoted by ( z1, z2, z3, z4)

Invariance: Let w1, w2, w3,w 4 be the images of four distinct points z1, z2, z3, z4 in the z-plane

under a Bilinear Transformation w = dcz

baz

(ad-bc 0 ) then

(w1, w2, w3,w 4) = ( z1, z2, z3, z4)

The Transformation ))((

))((

321

321

wwww

wwww

=

))((

))((

321

321

zzzz

zzzz

is Bilinear Transformation which

transforms three distinct points into three specified distinct points.

Objective Questions:

1. Cauchy’s residue theorem states that ( )

A) ( ) 2c

f z dz i ( sum of the residues) B) ( ) 2c

f z dz i (sum of the residues)

C) ( ) 2c

f z dz ( sum of the residues) D) ( )c

f z dz i (sum of the residues)

2. 2

0

cos ,sinf d

( )

A) 1 1

,2 2

C

z z dzz zfi iz

B)

1 1

,2 2

C

z z dzz zfi z

C)

11 ,

2C

z dzzf zz i iz

D)

1 1

,2 2

C

z z dzz zfiz

3. The transformation that preserves the magnitude but not sense is called ( )

A) isogonal B)mapping C)non-conformal D)conformal

4. The mapping 1

wz

is called ( )

A) rotation B)translation C)expansion D)inversion

5. The fixed points of 6 9z

z

are ( )

A) 1,-1 B) i,-i C) i,2i D) -1,i

6.0

sin mxdx

x

= ____________________________

7. The transformation az b

wcz d

is bilinear if _____________

8. The number of fixed points in w a z are _________________

9. If 4 1 2 2

dx

x

,then

4

01

dx

x

______________________

10. Residue of cos z

z at 0z is ________________

UNIT-IV

(Fourier Series and Transforms)

Periodic Function: A Function f(x) is said to be Periodic with Period T (>0) if for all real x,

f(x+T)=f(x) and T is the least of such values.

Euler’s formulae: The Fourier series for the function f(x) in the interval , 2c c is given by

0

1

cos sin2

n n

n

af x a nx b nx

where

2

0

2

2

1

1cos

1sin

c

c

c

n

c

c

n

c

a f x dx

a f x nxdx

b f x nxdx

The values of 0 , ,n na a b are known as Euler’s Formulae.

Dirichlet Conditions: A Function f(x) has a valid fourier series expansion of the form

0

1

cos sin2

n n

n

aa nx b nx

provided:

(i) f(x) is well defined,periodic,single - valued and finite.

(ii) f(x) has a finite number of discontuities in any one period.

(iii) f(x) has at most a finite number of maxima and minima in the interval of definition.

Fourier series for Even and Odd Functions:

If f(x) is defined in , and f(x) is an even function, f(x) can be expanded as a series in the

form

0

1

cos2

n

n

af x a nx

Where

0

0

0

2

2cosn

a f x dx

a f x nxdx

If f(x) is defined in , and f(x) is an odd function, f(x) can be expanded as a series in the

form

1

sinn

n

f x b nx

where

0

2sinnb f x nxdx

Half range fourier series:

The Half range sine series in (0, ) is given by

1

sinn

n

f x b nx

where

0

2sinnb f x nxdx

The Half range cosine series in (0, ) is given by

0

1

cos2

n

n

af x a nx

Where

0

0

0

2

2cosn

a f x dx

a f x nxdx

Fourier series for functions having period 2l:

The Fourier series for the function f(x) in the interval , 2c c l is given by

0

1

cos sin2

n n

n

a n x n xf x a b

l l

where

2

0

2

2

1

1cos

1sin

c l

c

c l

n

c

c l

n

c

a f x dxl

n xa f x dx

l l

n xb f x dx

l l

Fourier series for Even and Odd Functions:

If f(x) is defined in ,l l and f(x) is an even function, f(x) can be expanded as a series in the

form

0

1

cos2

n

n

a n xf x a

l

Where

0

0

0

2

2cos

l

l

n

a f x dxl

n xa f x dx

l l

If f(x) is defined in ,l l and f(x) is an odd function, f(x) can be expanded as a series in the

form

1

sinn

n

n xf x b

l

where

0

2sin

l

n

n xb f x dx

l l

Fourier Transforms:

Fourier integral theorem:

It States that 0

1cosf x f t p t x dtdp

Fourier sine integral:

0 0

2sin sinf x px f t ptdtdp

Fourier cosine integral:

0 0

2cos cosf x px f t ptdtdp

The infinite fourier transform of f(x):

The fourier transform of a function f(x) is given by F ipxf x F p f x e dx

The inverse fourier transform of F(p) is given by 1

( )2

ipxf x F p e dp

Fourier Sine Transform:

0

sins sF f x F p f x pxdx

0

2( ) sinsf x F p pxdp

Fourier Cosine Transform:

0

cosc cF f x F p f x pxdx

0

2( ) coscf x F p pxdp

Objective Questions:

1. If f(x) is an even function in (- ), then the graph of f(x) is symmetrical about the

________________

2. Fourier series expansion of an odd function in (-l, l) has only ______________ terms

3. A function f(x) defined for 0<x< can be extended to an odd periodic function in (- )

such that f(-x) = __________________

4. The rate of convergence of a Fourier series increases while the series is

___________________

5. If f(x) =

then f(x) is an ____________ function in (-1, 1)

6. If the Fourier cosine transform of e-ax

is

then fourier transform of

x e-ax

is ( )

a)

b)

c)

d)

7. If Fc (f(x)) = e-s²/4a²

/a then, Fc(x (f(x))) = ( )

a)

e

-s²/4a² b)

e

-s²/4a² c)

e

-s²/4a² d)

e

-s²/4a²

8. If the Fourier transform of f(x) Fs(f) =

, then F[f(x-a)] = ( )

a)

b)

c) e

ias

d) e

-is

9. If the Fourier transform of f(x) =

then fourier transform of f(ax) =

( )

a)

b)

c)

d)

10. The finite Fourier sine transform of x in (0, ) is ( )

a)

b)

c) (-1)

n d) (-1)

n+1

UNIT-V

(Applications of Partial differential equations)

Classification of Second order partial differential equations:

A partial differential equation of the form ( , , , , ) 0xx xy yy x yAu Bu Cu F x y u u u is said to be

(i) Elliptic: if 2 4 0B AC at a point in the (x,y) plane (Laplace equation)

(ii)Parabolic: if 2 4 0B AC at a point in the (x,y) plane (Heat equation)

(iii)Hyperbolic: if 2 4 0B AC at a point in the (x,y) plane (Wave equation)

Method of Separation of variables

A partial differential equation involving two independent variables say x and y,we seek a

solution in the form X(x).Y(y) and write down various types of solutions.

One Dimensional wave equation

2 2

2 2 2

1y y

x c t

Where 2c T m with T= tension in the string at any point and m is mass per unit length of the

string.

Subject to the conditions

0, 0

, 0

y t foralltEndconditions

y l t forallt

,0 ,0

,0t o

y x f x x l

Initialconditionsyg x x l

t

One Dimensional Heat Equation:

2

2 2

1u u

x c t

where 2 k

cs

is called the diffusivity of the substance

k is thermal conductivity, is density and s is specific heat

Objective Questions:

1. The nature of the PDE 2 2

2

2 2

z zc

y x

is ( )

a)hyperbolic b)parabolic c)elliptic d)circle

2. The PDE 2 2 2

2 2,

u u u u uA B C D E Fu f x y

x x y y x y

( )

a) 2 4 0B AC b) 2 4 0B AC c) 2 4 0B AC d) 2 4 0B DC

3. One dimensional wave equation is given by ( )

a)2 2

2

2 2

z zc

y x

b)

22

2

z zc

y x

c)

2 2

2 20

z z

y x

d)

2 2 2

2 2 20

u u u

y x z

4. One dimensional heat equation is given by ( )

a)2 2

2

2 2

z zc

y x

b)

22

2

z zc

y x

c)

2 2

2 20

z z

y x

d)

2 2 2

2 2 20

u u u

y x z

5. The equation for the PDE 4 3u u

ux y

by variable separable method is ( )

a) ' ' 3

4X Y Y

kX Y

b)

' ' 34

X Y Yk

X Y

c)' ' 3X Y Y

kX Y

d)

' '3

X Yx y k

X Y

6. The nature of the PDE 2 2

2 2

2 21 0 1

u ux y ify

x y

is ________________________

7. The Solution of 3 2 0, 0, 4 yu uu y e

x y

is ____________________________

8. The possible boundary conditions for wave equation are _________________________

9. The solution of heat equation with the boundary conditions 0, 0, , 0u y u l x and

,0u x f x is ________________________

10.The variable separable form of the equation 2 2

2

2 20

u u ur r

r r

is

_____________________