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复变函数积分

\int_{|z|=r} \operatorname{Re} z d z

Sep 25, 2024

复变函数积分

Solution by Steps

step 1

The integral we need to evaluate is given by

I = \int_{|z|=r} \operatorname{Re} z \, dz

step 2

We can express

\operatorname{Re} z

in terms of

z

\operatorname{Re} z = \frac{z + \overline{z}}{2}

step 3

Thus, the integral becomes

I = \int_{|z|=r} \frac{z + \overline{z}}{2} \, dz

step 4

Since

\overline{z} = \frac{r^2}{z}

on the contour

|z|=r

, we can rewrite the integral as

I = \int_{|z|=r} \frac{z + \frac{r^2}{z}}{2} \, dz

step 5

This simplifies to

I = \frac{1}{2} \left( \int_{|z|=r} z \, dz + \int_{|z|=r} \frac{r^2}{z} \, dz \right)

step 6

The integral

\int_{|z|=r} z \, dz = 0

(by Cauchy's integral theorem) and

\int_{|z|=r} \frac{r^2}{z} \, dz = 2\pi i r^2

step 7

Therefore, we have

I = \frac{1}{2} \left( 0 + 2\pi i r^2 \right) = \pi i r^2

Answer

The value of the integral is

\pi i r^2

Key Concept

The integral of a complex function over a closed contour can often be evaluated using properties of analytic functions and Cauchy's integral theorem.

Explanation

In this case, the integral evaluates to zero for the first term due to the symmetry of the contour, while the second term contributes a non-zero value, leading to the final result.