3.6 Alternative Integral Formulas

As in the Cauchy Integral Formula (Theorem 3.1.8), we can write holomorphic functions in terms of an integral representation. We define the Cauchy kernel to be

Then Theorem 3.1.8 can be written as

There also exist other integral formulas for functions, varying in the kernel of the expression.

Let be harmonic such that is continuous on . By the mean-value property introduced in Lemma 3.4.1, we have

where . By the uniform continuity of on (Theorem 1.2.15), , such that for all satisfying and all ,

It then follows that

Hence,

Let and notice that

maps to bijectively. Let be harmonic on and continuous on . Then is also harmonic on , and by (3.6.1),

By the univalence of , let . It follows that

Then from (3.6.1),

Let

known as the Poisson kernel. Then,

where . (3.6.3) is also known as the Poisson Integral Formula.

For all , where , we can apply the transformation

to extend the automorphism to . Let instead be harmonic on and continuous on . Then,

It follows that is also harmonic on with

and from the bijectivity of ,

Then because ,

The expression

is a general form of the Poisson kernel. Then with ,

The Poisson kernel can also be rewritten as

Thus, (3.6.5) is equivalent to

Since , , and

where . Since and for all and , the function

is holomorphic on : for each fixed , the integrand is holomorphic in , and on compact subsets of we may differentiate under the integral sign. Therefore, is the real part of a holomorphic function

where . Since is holomorphic, by Proposition 2.2.1, is constant. For ,

Letting , the integral vanishes, and we obtain .

Define the Schwarz kernel to be

Then for a holomorphic function on that is continuous on , we obtain the Schwarz Integral Formula:

The significance of this alternative formula implies that a holomorphic function can be recovered from the real part on the boundary of a disk and the imaginary part at a single point.

From (3.6.7), we can rewrite

Let

which is known as the conjugate Poisson kernel. Then (3.6.7) yields yet another integral representation of harmonic functions:

where . Two harmonic functions are said to be conjugate if they are the real and imaginary parts of a holomorphic function. As seen above, on open disks, any harmonic function will admit a unique conjugate, up to an additive constant . For a harmonic function , we can construct its harmonic conjugate from (3.6.9).

The Poisson kernel is important in many branches of mathematics. We will introduce two of the important uses below.