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Lecture 19

Cauchy’s Integral Formula

19.1 Consequences of Cauchy’s Integral Theorem

Theorem 19.1 In a simply connected domain, an analytic function has an antiderivative,

its contour integrals are independent of path, and its loop integrals vanish.

Given that f (z) is analytic inside and on the simple closed contour Γ, we know from CITthat

Γ f (z)dz = 0. However, what about,

Γ

f (z)

(z − zo)dz,

where zo is a point in the interior of Γ?Your first guess might be that this is zero? But based on our previous experience there isno reason to believe this. Another guess would be that the integral is proportional to f (zo),which we shall show now with Cauchy’s Integral Formula

Theorem 19.2 Let Γ be a simple closed positively oriented contour. If f (z) is analytic in

some simply connected domain D containing Γ, and zo is any point inside Γ, then

f (zo) =1

2πi

Γ

f (z)

z − zodz (19.1)

Proof: We first note that f (z)/z − zo is analytic everywhere in D except for the point zo.Therefore, by DIT, the integral over Γ can be equated to the integral over some positivelyoriented circle, C r : |z − zo| = r,

Γ

f (z)

z − zodz =

C r

f (z)

z − zodz

But we may rewrite this as,

C r

f (z)

z − zodz =

C r

f (zo)

z − zodz +

C r

f (z)− f (zo)

z − zodz

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LECTURE 19. CAUCHY’S INTEGRAL FORMULA 83

The first integral is trivially,

C r

f (zo)z − zo

dz = f (zo) C r

1z − zo

dz = 2πif (zo)

which is the answer we sought. Thus we have,

Γ

f (z)

z − zodz = 2πif (zo) +

C r

f (z)− f (zo)

z − zodz

Note that the 1sttwo terms above are independent of radius r, therefore the last term shouldnot change upon letting r → 0.

limr→0+

C r

f (z)− f (zo)z − zo

dz

We thus see that Cauchy’s Integral formula will follow if the above equation is zero! Let’sset M r = max[|f (z)− f (zo)|] therefore,

f (z)− f (zo)

z − zo

=|f (z)− f (zo)|

r≤

M rr

Thus, our integral becomes,

C r

f (z)− f (zo)

z − zo ≤ M rl(C r)

r=

M r2πr

r= M r2π

But since this must be continous at zo,

limr→0+

M r = 0

Which implies that,

limr→0+

C r

f (z)− f (zo)

z − zodz = 0

and we obtain Cauchy’s Integral Formula.

19.2 Consequences of Cauchy’s Integral Formula

Let us replace z by η and zo by z in CIF, then

f (z) =1

2πi

Γ

f (η)

η − zdη (z inside Γ)

This suggests a formula for the derivative f (z),

f (z) =1

2πi Γ

f (η)

(η − z)2dη (z inside Γ)

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Theorem 19.3 Let g be continous on the contour Γ, and for each z not on Γ set:

G(z) ≡ Γ

g(η)η − z

dη

Then G is analytic and its derivative is:

G(z) =

Γ

g(η)

(η − z)2dη, for all z not on Γ. (19.2)

Observations:

(i) have not assumed that Γ is closed

(ii) have not assumed that g is analytic.

More generally we have,

H (z) =

Γ

g(η)

(η − z)2dη (z not on Γ)

H (z) = 2

Γ

g(η)

(η − z)3dη (z not on Γ) (19.3)

What does this imply: Derivative of an analytic function is again analytic:

Theorem 19.4 If f (z) is analytic in a domain D, then all of its derivatives, f

, f

, . . . , f n

,exist and are analytic in D.

Note: This is not true for real functions, consider f (x) = x5/3,

f (x) =5

3x2/3

does not have a derviative at x = 0.

19.3 Summary

Analyticity of derivatives implies that, given a function f (z) which has an antiderivative ina domain D, then f itself must be analytic in D. But by one of our theorems, the existenceof an antiderivative for a continuous function is equivalent to the property that all loopintegrals vanish and, therefore, we arrive at Morea’s Theorem:

Theorem 19.5 If f (z) is continous in a domain D and if,

Γ

f (z)dz = 0

for every closed contour Γ in D, then f (z) is analytic in D.

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Finally, as a consequence of all of the above, we arrive at the generalized Cauchy Integral

Formula:

Theorem 19.6 If f is analytic inside and on the simple closed positively oriented contour

Γ and if z is any point inside Γ, then

f (n)(z) =n!

2πi

Γ

f (η)

(η − z)n+1dη, (n = 1, 2, 3, . . .) (19.4)

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