further generalizations of the cauchy integral formula

Further Generalizations of the Cauchy Integral FormulaAuthor(s): Dawson G. FultonSource: American Journal of Mathematics, Vol. 61, No. 4 (Oct., 1939), pp. 843-852Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371630 .

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FURTHER GENERALIZATIONS OF THE CAUCHY INTEGRAL FORMULA.*'

By DAWSON G. FULTON.2

1. The generalizations given in this paper arise out of a consideration of the structure of the Cauchy Integral Formula when it is freed of the special form in which it appears due to the use of complex numbers. Thus in the formula

1 f(z) dz. (1)~~~~ ~ t(a) 27r z -' a

we set a O, f(z) =Y + iX, z=x + iy, dz dx + idy, -i(dx + idy) = (a + If) ds, where a and / are the direction cosines of the normal to the contour; then by equating real and imaginary parts we get

27rXo-c x xar + Y, +Y ' }( ds

2 Y2=f {X( 9 x+Y8 tY Y}ds, 2 = 2 + y2)

We notice that the integrand is linear and homogeneous in the components X and Y, and also in the direction cosines a and ,6, and homogeneous of degree -1 in the independent variables x and y. More specifically we say that the integrand is linear a-id homogeneous in the independent variables besides containing in the denominator a quadratic form of the independent variables.

We choose these observations as the starting point of our generalizations, and propose the following considerations. Is it possible, and, if so, under what conditions, to find a general integral formula of this same structure which can be used for the same purpose as is the Cauchy Integral Formula; that is, for the calculations of the values of functions in the interior points of a region when their values are given on the boundary? Using the index notation, denoting X by XI, Y by X2, x by x1, y by x2, etc., summations by Greek letters used as subscripts and superscripts, we write the integral we wish to consider as

* Received November 28, 1938. 'Presented before the American Mathematical Society, Chicago, April, 1937. 2 The writer wishes to express his appreciation to G. Y. Rainich, of the University

of Michigan, for his helpful suggestions during the preparation of this paper.

843



844 DAWSON G. FULTON.

(3) 5' r Nix xaXxavds,

where the constants NiJ are arbitrarily chosen, and r2 =

g"apX5Xx is a positive kl

definite form. The original formula (2) is obtained from this by setting

(4) = 8*lSjk + Siil - a*jl; gij =8j,

where Sij etc. mean the Kronecker delta. &*j wherever used throughout the paper will have the same meaning.

The formulas (2) are applicable to analytic functions; hence in our consideration to a pair of functions X and Y, which are differentiable (possess Stolz differentials) and satisfy the conditions-the Cauchy-Riemann differential equations-

ax, OX2 Ox'- OX2 81+ OX2 O, X2 0.2

The proof that the formulas (2) hold for functions which satisfy (5) is based on the following points: (1) that the integrals are independent of the particular contour about which they are taken as long as they surround the point at which we wish to make the evaluation, (2) that calculations are made of integrals over a circle around the point in question, and (3) that the desired evaluation is obtained by passing to the limit by making the radius of the circle go to zero. We want to know what conditions must be imposed on the general formula (3) so that the same results can be obtained.

2. In discussing these conditions let us consider the special case where g=j 8*j. First of all we rewrite formula (3) in a form which will evaluate the functions X and Y at any inner point a (a', a2), rather than at the origin. Formula (3) then becomes

(6) (X')a = 2 j Niv (xa-aa)Xxvds,

where by (X)a we mean X evaluated at the point a, and

r2 (Xa - aaf) (xa - aa).

Since the integral is to be independent of the contour about which it is taken, we know that

(7) a (coefficient of av) =0;



GENERALIZATIONS OF THE CAUCHY INTEGRAL FORMULA. 845

that is, ox x

(xa_ aa) (xa - a) (x6-a)a) N * h + (Xa aa)(Xa aa)NiaXx

2 (x? ao) (xp- aP)N i^x = 0.

This equation must be true for all points a; hence it will hold for ta where t takes on all values such that ta is a point within R. This will give a cubic equation in t which must be true identically; that is, the coefficient of each term must vanish. Writing down the coefficient of t3, we get

axx aaaaa6NiP ~~ 0. C AaXv ?'x

This equation must be true identically in a, thus

axx (8) NipV -X 0

Applying this formula (8) to the term which is free of t; namely,

XaXaX?Niv a + XaxaNio XX - 2xfxlNiv Xx =0,

we get xaXaNia XX - 2xax'N'v Xx = 0.

This equation holds for all X* and xi; hence we get

(9) 2jANa - 2 (Nil + N*k)

By an easy substitution we find that for j= k 1 or 2, N1' - N;2 and lit 21

that for j 7#e, N = Ni'. The conditions (8) and (9) assure the kl it

vanishing of the coefficients of t2 and t. Thus we see that (8) and (9) are sufficient conditions that the integral formula (6) be independent of the contour.

Let us assume the conlditioils (8) and (9) and then we may integrate the formulas (6) about any contour whatsoever. We shall choose a circle with center at the given point a. Then

(10) (Xi a')/r (X (x2-a2)/rj.

Using (9) and (10) the formulas (6) become

1 0 ~~~~~ds (Xi)a4= - (NilXl +Nil X2)

But this integral is independent of r, and we may write




lf X(Nil '+Nt X2) ds Nl (X')a + Nil (2)a.

Since this must give the required evaluation (Xi) a, we see that when i = 1

(Xl) a- NJ(Xl) a + N" (X2)a.

Hence N1 -1 and N" = 0. Similarlv when i = 2 N 21 =0 N2t = 1. 11 12 11 12

From these values, together with formulas (9), it is easily verified that

(11) N? =

il

and therefore (12) NkJ + Nik 8jkNi= 28jk8il

ki it at

Thus we see that (8) and (12) are sufficient conditions which must be imposed upon the functions X and Y and the constants Nii in order to

kl

insure the fact that the formulas (6) will evaluate the given functions at any interior point of a region in terms of the boundary values of the functions.

3. One other point must be settled, however, before we state the theorem. The system of differential equations (8) consists of four linear homogeneous equations with constant coefficients. In order for this system to have a solu- tion other than the trivial case where the functions are all constants, the determinant of the matrix Nii must be equal to zero. This means, of course, that there must be some relationship among the equations (8), and suggests that it might be possible to find an equivalent set of fewer equations. The number of this latter set will, of course, depend upon the rank of the matrix Nkl. If one writes out the matrix using (11) and (12) it is obvious that the rank cannot be one unless we use pure imaginaries for the constants Ni and we do not wish to do this. Since the equations do not reduce to one equation, then, let us look at the next most interesting case; namely, when

the rank of the matrix is 2. If we write out the first two equations for which i= 1 we find

(13) ox + (qX + sY) ax _ (qX + sY) O

12 = -N a =0, a

where N'2 N"1 q, N12 -IN = s. These are the Cauchy- 11 21 12 22

Itiemann differential equations for the functions X and qX + sY. Because

of these equations then we may write integral formulas for X and qX + sY exactly as we did for X and Y in (2). Thus we find (X)a and (Y)a as a

result of the first two equations only. That is, while the four equations in (8) are sufficient, the two for which i = 1 or 2 are all which are necessary for




the possibility of the desired evaluations. We note also that for this set-up either of the two pairs of equations will also insure the independence conditions which we discussed in section 2. Thus we see that the set of equations (8) breaks up iilto two sets of two equations (those obtainled by setting i = 1, or those for which i = 2), either one of which is sufficient for our purposes. If we have one of these pairs of equations given alld we wish to evaluate both (X)a and (Y)a by using formula (6) only, then we shall have to make the two pairs of equations equivalent. This means, as is readily checked, that we impose the further condition on the constants

1. + (Nii)2 (14) NS. -- Ni., (i74i) = , (i j1).

This is quite similar to the reduction of the equations (8) when we give to N4j the evaluation (4). Using this evaluation we get two sets of two equa-

kt

tions each, and the sets are identical. In the case which we have discussed in this section one set may be transformed into the other by linear transformations on the functions onlly.

We now state the followinlg theorem:

Given any r-egiont R bountded by C, the functions X and Y may be evaluated at any inner point ai of R in teirms of their values on C by means of the integral formulas

(X )a fi Nil (xO' aa) cvd; 2- ( aa)(x aa) (XY t) a = j .2 ax ,( z- ) Yla,cds; 2=( xc _ a (xa _a

provided the functions X and Y possess a Stolz differential and satisfy within thte regiomI R the system of differential equations

axx kZ. X Ox,

and the constants N/i satisfy the conditions

Ni + Nk =28jk8il,

and for i=/=j

= -NS i 1 + (IV =i)2 ii i N~~iVI jii

4. In sections 2 and 3 we have demonstrated the conditions of which we inquired in section 1, but so far we have consideTed only the special case where

-.2 (xa - aa) (xa - aa).




We wish now to discuss the more general expression, for (6) when we use for r2 the positive definite form

9a8(-za aa) (Xl a/3)

We wish to find the conditions which we must impose in this case. If we carry through the same procedure here as we did in section 2, we find the conditions for independence of contour to be that the functions must satisfy

( 15 ) Aav ? where

gvjlVN + gvkNiP gikNt.

These and subsequent conditions could be obtained immediately from the results of sections 2 and 3 if we could find a linear transformation oni the variables which would carry the form (x" - aa) (xa - al) into

aO(xa - aa) (xO - a) for any given- set of gij's. Such a transformation does exist, we know, and is found by choosing as the constants c,j in the transformation the components of two vectors the square of whose lengths are the positive numbers g11, and g22, g12 being the scalar product of the two; that is, gij = cxicxj. Let us apply this transformation, xt - c'p (xp) (x-)'- cpxP, to the variables in the formulas (8). We get

(17) j 2 cPac',vNir(x" aa)X cavds,

which we write as

f4 Mip (x- a) X, vds,

where

MIk cpoC'rjNpt; ='- ga( ax ) (xa -MXr); gij =-cxCxj.

But by (16) we know that

g Ii za= gv~ jMh + gVkM.

That is, CXjCXkMli = CXvCXj1M' + CXvCXkMAi"

Therefore CXpC jPCXTCkTrM = CjpCIkCvCxvCxpM> + C jpC 'k7CVCxTvMi v

or

8jklli6 c?.cpMvAlil + c'jPckv,7M1, since c jpC,kp S= a

But Mia=f Ni', and c'j,ckviAll Ni'.

Helice we get Sjk.Xi l N ik + N.i . ayt it k I




But this is exactly equation (9) found in section 2. Thus we have shown that this general condition is derivable from the special case of section 2 and 3 by means of linear transformations oni the variables. It is therefore possible to apply the reasoning of those sections to this general case. We get then

(18) gvjN'v + gweNv --gjN N 2gjv8i

as the conditions to be satisfied by the constants N1J and gij for this case. This brings us then to the following theorem:

Given any region R bounded by C, the functions X and Y may be evaluated at any inner point a of R in term,s of their values on C by rmeans of the integral formulas

N iv N(xo- a6)XXvds; r2 ga(Xa aa) (x M),

provided the functions X and Y possess a Stolz differential and satisfy within the region B the system of differential equations

kX Ox"

and the constants N and g satisfy the conditions

gvi N4 + gVkMN =-- 29g jk8 .

Here as before the determinant of the matrix NI'i must be zero. The k I additional conditions necessary to have an equivalent set of two equations are not developed here.

To verify the fact that this does give the desired values, one integrates about the ellipse ga (xa - aa) (x- a) K.

5. Instead of starting with ani iiitegral representation, it is possible to approach this problem by considering two differential equations of the form

A -X+B X+0C a + D ,= O (19)

Ox aaT Y ax1 ax d dY

A'- + B'- + C' a + D' a =0. TX ay ax T

This system was treated by H. C. Chang in his thesis on " Transformations of Linear Partial Differential Equations." 3 He showed that the general function X + iY could be reduced to an analytic function when X and Y

See H. C. Chang, " Transformation of partial differential equations," Science Society of China, vol. 7 (1931), no. 2.




satisfied the system of differential equations (19), provided the functions X and Y are differentiable, and that the following condition on the coefficients of (19) is satisfied

(20) 4(BD' B'D) (AC'- AOC) - (AD' A'D + BCO -BC)2 > 0.-

We ask now whether it is possible to find an integral formula analogous to the Cauchy Integral formula whereby we shall be able to evaluate this general functioll at some inner point of a region in terms of its values on the boundary. SuchI a formula was obtained by applying the same linear transformations to the functions and variables in the Cauchy Integral Formula as were applied to the Cauchy-Riemann Differential Equations to secure the system (19). The formula has as its constants certain expressions involving the coefficients of (19). These expressions are combinations of M,j's, where by Mij we mean the determinant formed by the i-th and the j-th columns of the matrix

A B C D (21) A' B' C' D'

As a conveiience in writing, the following lnotations have been introduced

(22.1) R2 4M13M24- (M14 + M23)2,

(22. 2) g M4 X2 (M14 + M123)xy + M13y2.

Using this notation we now state the following theor-em:

If R is, a given region, of vhich C is the boundary, then the functions X and Y may be evaluated at the origin by means of the formulas

(23. 1) 27rXo = If [ { (M14- M23) X-2M34Y} dg + R2 (Xa + y3) XdS],

(23. 2) 27rYo=S [ {2M12X + (M14-M23) Y} d g-R 2 (X: + y3) Yds]

provided the functions X and Y are differentiable and satisfy the differential equations

ox ax ai ai A 0Z+ B a + C ady+ D dgy-=O ox ax Oxy a

A' ax + B' ox

+ C'ay + D'0aY =

whose coefficients satisfy the inequality (20).

Obviously a translation of the axes will make this formula applicable to any inner poilit of the region considered.




The integral formulas (23), together with the differential equations (19), are derived in the same manner as were the corresponding formulas and equations in section 4; namely, by means of linear transformations. In this case we have two linear differential equations instead of four. This, then, is clearly a special case of that considered in section 4.

6. The discussions of sections 2, 3 and 4 dealt entirely with the case of two dimensions. Integral formulas which are generalizations of the Cauchy Integral Formula for analytic functions of higher dimensions have been obtained, and the results appear in a joint paper by G. Y. Rainich and the writer.4

We wish to consider now the extension within these higher dimensions similar to the one which we have given for two dimensions. For the general case of n dimensions we study the integral formulas

(24) )(X')a -Niv (x6-a6)XXcvdw, r( 1n) i'n0-/

where r2 = ga (xa - aa) (xs- a-3), and all summations are from 1 to n. We

want to know what conditions must be imposed here in order that we may evaluate the n functions at any inner point of an n-dimensional region in terms of their values on a hyper-surface which bounds the region. The argu- ments of the previous sections carry through here very readily, and we may state the following gelleral theorem:

If T is a given n-dimensional region bounzded by a hypersurface 8, then we may evaluate the functions Xt (i = 1, 2, n) at any inner point

: (a1, a2, . , an) in teirms of their values on the hypersurface by means of the integral formrulas

(25) P( l) (Xi) a f Ni (x? -a)YXavdw,

where 1-2 = g. (Xa - aa) (x a), provided the functions Xt possess Stolz differentials and satisfy withini the region T the following system of differentiable equations

axx (26) NPaxv 0

together with the conditions that

(27) n(gvjNip + gVkNiP) = 2gjvN' =- 2ngjk8- .

4 G. Y. Rainich and D. G. Fulton, "Generalizations of the Cauchy integral formula to higher dimensions," American Journal of Mathematics, vol. 54 (1932), pp. 235-242.




The formulas derived earlier in the paper are clearly special cases of this general set-up. By setting n = 2, we get the formulas and corresponding conditions of section 4, and for n 2 and gij = 8j we get the results of sections 2 and 3.

It might further be pointed out that the formulas given in the joint paper referred to above are also special cases of this general formula (25); they may be obtained by givilng to n the proper value, and to N7J and gij the

evaluation given in (4) section 1. Likewise formula (26), under the same substitution for N'J, yields the corresponding set of differential equations for

each case. It will be noted that for this special evaluation of the constants N'J the number of differential equations (26) reduces to (n2- n + 2)/2.

It should be pointed out that the reduction of the number of differential equations for the general case is not considered in this paper.

ARMOUR INSTITUTE OF TECHNOLOGY.



further generalizations of the cauchy integral formula

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