**for**

**6.5 Integral Representations
for Analytic Functions**

We now present some major results in the
theory of functions of a complex variable. The first
result is known as __Cauchy's__
__integral
formula__ and shows that the value of an analytic
function f(z) can be represented by a
certain contour integral. The derivative, , will
have a similar representation. In Section
7.2, we use the Cauchy integral formulas to prove __Taylor's__
__theorem__
and also establish the power series representation for analytic
functions. The Cauchy integral formulas are a convenient tool for
evaluating certain contour integrals.

**Theorem 6.10 (**__Cauchy
Integral
Formula__**).** Let
f(z) be analytic in the simply
connected domain D, and
let C be a simple closed positively
oriented contour that lies in D. If is
a point that lies interior to C,
then

.

**Proof of Theorem 6.10 is in the book.
**

**Example 6.21.** Show
that , where
C is the circle with
positive orientation.

Solution. We have and . The
point lies
interior to the circle, so Cauchy's integral formula implies
that

,

and multiplication by establishes
the desired result.

**Example 6.22.** Show
that , where
C is the circle with
positive orientation.

Solution. Here we have . We
manipulate the integral and use Cauchy's integral formula to
obtain

**Example 6.23.** Show
that , where
C is the circle with
positive orientation.

Solution. We see that . The
only zero of this expression that lies in the interior of
C is .

We set and
use Theorem 6.10 to conclude that

**Theorem 6.11 (**__Leibniz's
Rule__**).** Let
G be an open set, and
let be
an interval of real numbers. Let
and its partial derivative
with respect to z be continuous
functions for all z in G
and all t in I. Then

is
analytic for z in G, and

.

**Demonstration
for Theorem 6.11.**

We now generalize Theorem 6.10 to give an
integral representation for the
derivative, .
We use __Leibniz's__
__rule__
in the proof and note that this method of proof is a mnemonic device
for remembering Theorem 6.12.

**Theorem 6.12 (Cauchy's Integral Formulae
for Derivatives).** Let
be analytic in the simply connected domain D,
and let C be a simple closed
positively oriented contour that lies in D. If
z is a point that lies interior to
C, then for any integer ,
we have

.

**Proof of Theorem 6.12 is in the
book.
**

**Example
6.24.** Let denote
a fixed complex value. Show that, if C
is a simple closed positively oriented contour such
that lies
interior to C, then

, and

(6-50)

, for
any integer .

Solution. We let . Then for . Theorem 6.10 implies that the value of the first integral in Equations (6-50) is

,

and Theorem 6.12 further implies that

.

This result is the same as that proven earlier in Corollary 6.1. Obviously, though, the technique of using Theorems 6.10 and 6.12 is easier.

**Example 6.25.** Show
that, where
C is the circle with
positive orientation.

Solution. If we set , then
a straightforward calculation shows that . Using
Cauchy's integral formulas with ,
we conclude that

We now state two important corollaries of Theorem 6.12.

**Corollary 6.2.** If
is analytic in the domain D, then all
derivatives
exists
for (and
therefore are analytic in D).

**Proof of Corollary 6.2 is in the
book.
**

**Remark 6.3.** This
result is interesting, as it illustrates a big difference between
real and complex functions. A real function
can have the property that
exists everywhere in a domain D, but
exists nowhere. Corollary 6.2 states that if a complex
function
has the property that exists
everywhere in a domain D, then,
remarkably, all derivatives of
exist in D.

**Corollary 6.3.** If
is a harmonic function at each point
in the domain D, then all partial
derivatives ,
,
,
,
exists
and are harmonic functions.

**Proof of Corollary 6.3 is in the
book.
**

**Extra Example
1.** Show that the partial derivatives
of
are harmonic functions.

**Exercises
for Section 6.5. Integral Representations for Analytic
Functions**

**The Next Module
is**

**Theorems
of Morera and Liouville and Extensions**

**Return to the Complex
Analysis Modules**

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell