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## C. Results from Analytic Function Theory

#### C.8.1 Poisson's Integral for the Half-Plane

Theorem C.9  Consider a contour bounding a region . is a clockwise contour composed by the imaginary axis and a semicircle to the right, centered at the origin and having radius . This contour is shown in Figure C.4. Consider some with .

Let be a real function of , analytic inside and of at least the order of ; satisfies

 (C.8.1)

then

 (C.8.2)

Moreover, if (C.8.1) is replaced by the weaker condition

 (C.8.3)

then

 (C.8.4)

##### Proof

Applying Theorem C.8, we have

 (C.8.5)

Now, if satisfies (C.8.1), it behaves like for large , i.e., is like . The integral along then vanishes and the result (C.8.2) follows.

To prove (C.8.4) when satisfies (C.8.3), we first consider , the image of through the imaginary axis, i.e., . Then is analytic inside , and, on applying Theorem C.7, we have that

 (C.8.6)

By combining equations (C.8.5) and (C.8.6), we obtain

 (C.8.7)

Because , the integral over can be decomposed into the integral along the imaginary axis , , and the integral along the semicircle of infinite radius, . Because satisfies (C.8.3), this second integral vanishes, because the factor is of order at .

Then

 (C.8.8)

The result follows upon replacing and by their real; and imaginary-part decompositions.

Remark C.1  One of the functions that satisfies (C.8.3) but does not satisfy (C.8.1) is , where is a rational function of relative degree . We notice that, in this case,

 (C.8.9)

where is a finite constant and is an angle in .

Remark C.2   Equation (C.8.4) equates two complex quantities. Thus, it also applies independently to their real and imaginary parts. In particular,

 (C.8.10)

This observation is relevant to many interesting cases. For instance, when is as in remark C.1,

 (C.8.11)

For this particular case, and assuming that is a real function of , and that , we have that (C.8.10) becomes

 (C.8.12)

where we have used the conjugate symmetry of .

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