Annals of Mathematics

Note on Cauchy's Integral FormulaAuthor(s): J. L. WalshSource: Annals of Mathematics, Second Series, Vol. 18, No. 2 (Dec., 1916), pp. 79-80Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/2007173 .

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NOTE ON CAUCHY'S INTEGRAL FORMULA.

BY J. L. WALSH.

It is the object of this note to establish Cauchy's Integral Formula by means of the theorem* which is usually called " the mean value theorem for harmonic functions," but which can also be called the " mean value the- orem for conjugate functions." Two functions u(x, y) and v(x, y) are said to be conjugate in a region T of the x, y-plane if throughout this region they are single-valued, continuous, and have continuous first partial de- rivatives, and also

au _ v au av (1) ax = y' ay ax

If u and v are conjugate functions in a simply connected region T, it is shown below without assuming the existence of second derivativest that

(2) 2 u(x, y)ds = u(xo, yo),

(3) Xfv(x, y)ds v(xo, yo),

where the integrals are extended around any circle lying wholly within T, the radius of the circle being denoted by r and the center by (xo, yo).

Suppose that, within T, f(z) = u(x, y) + iv(x, y) is an analytic and single-valued function of z, where z = x + iy. Then u and v are con- jugate functions throughout T, so that equations (2) and (3) are true, the integrals being taken around the circle described above. Multiplying (3) by i and adding it to (2), we have

(4) 1 f(z)ds = f(zo), zo = xO + iyo.

If z is on the circle, the use of polar coordinates gives

z - zo = r(cos so + i sin p), * This is the reverse of the procedure common in books on Theory of Functions. For ex-

ample, in Burkhardt, Funktionentheorie, ? 36, the mean value theorem for harmonic functions is derived from Cauchy's Integral Formula by the use of essentially the same analytic machinery as that employed here.

t It is, of course, true that u and v have continuous second partial derivatives; but we must not assume that fact at this point if we wish to prove it later by means of Cauchy's integral formula, as is commonly done.

79

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80 J. L. WALSH.

dz = r(- sin so + i cos p)dp,

i(z -Zo) d r

Substituting the value of ds in (4), we have

1 J (Z) dz f(zo),

where the integral is taken around a circle whose center is zo. As the formula is true for this particular path surrounding zo, it is true, by Cauchy's theorem, for any path within T surrounding zo once in the positive direction.

The proof* of formulas (2) and (3) is as follows. Using polar co6rdi- nates (p, jo) with the pole at (xo, yo),

au au Co o u si.p qa = acos 5o+ qysrn so, 1 v av ir\ dv

ax cos (o + 2 J+ j sln (s +

au Op

by equations (1). Then W2 au 1 (Ou 1 r'iav 1 fOv d=- 5- ds =J- - ds=- -ds=O, Jo aP PJ Op p p 5, p, OS

all the integrals being taken around a circle whose radius is p and center (xo, yo). Hence

=f0 dpf 2 dup = dqpf, tdp = f [u(x, y) - u(xo, yo)]d9p;

* u(xO, Yo) u(x, y)dfp

= 2wrfJ u(x, y)ds. Similarly

V(XO, YG) = 2rr Vx, y)ds,

the integrals being taken around a circle lying wholly within T, the radius of the circle being r and the center (xO, yo).

HARVARD UNIVERSITY, CAMBRIDGE, MASS.,

April, 1916. * This proof is the proof given by M. Bocher, "Gauss's Third Proof of the Fundamental

Theorem of Algebra," Bull. Amer. Math. Society, vol. I (Second Series) (1894-95), pages 206, 207.

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