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Question

Math 6720 • Applied Complex Variables and Asymptotic Methods

Summer 2022 • Question 1

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods Summer 2022 • Question 1

Let \(C\) be the unit circle centered at the origin. Cauchy’s integral formula implies that \[ f(z)=\frac {1}{2\pi i}\oint _C \frac {f(\xi )}{\xi -z}\,d\xi ,\qquad |z|<1. \]

Perform the change of variables \(\xi =e^{i\theta }\) to show that \[ f(z)=\frac {1}{2\pi }\int _0^{2\pi }\frac {f(\xi )\xi }{\xi -z}\,d\theta , \] where \(z\) lies inside the unit circle. Explain why \[ 0=\frac {1}{2\pi }\int _0^{2\pi }\frac {f(\xi )\xi }{\xi -1/\bar {z}}\,d\theta . \]
Using the fact that \(\xi =1/\bar {\xi }\) on the unit circle, show that \[ f(z)=\frac {1}{2\pi }\int _0^{2\pi } f(\xi ) \left (\frac {\xi }{\xi -z}\pm \frac {\bar {z}}{\bar {\xi }-\bar {z}}\right )\,d\theta . \]
Taking the plus sign in part (b), deduce the Poisson formula for the real part \(u(r,\phi )\) of \(f(z)\) with \(z=re^{i\phi }\): \[ u(r,\phi )=\frac {1}{2\pi }\int _0^{2\pi } \frac {1-r^2}{1-2r\cos (\phi -\theta )+r^2}u(1,\theta )\,d\theta . \] This shows that the real part of \(f(z)\) on the unit circle determines the real part of \(f(z)\) everywhere inside the unit circle.

Let \(C\) be the unit circle centered at the origin. Cauchy's integral formula implies that
\[
f(z)=\frac{1}{2\pi i}\oint_C \frac{f(\xi)}{\xi-z}\,d\xi,\qquad |z|<1.
\]
\begin{enumerate}
\item
Perform the change of variables \(\xi=e^{i\theta}\) to show that
\[
f(z)=\frac{1}{2\pi}\int_0^{2\pi}\frac{f(\xi)\xi}{\xi-z}\,d\theta,
\]
where \(z\) lies inside the unit circle. Explain why
\[
0=\frac{1}{2\pi}\int_0^{2\pi}\frac{f(\xi)\xi}{\xi-1/\bar{z}}\,d\theta.
\]

\item
Using the fact that \(\xi=1/\bar{\xi}\) on the unit circle, show that
\[
f(z)=\frac{1}{2\pi}\int_0^{2\pi} f(\xi)
\left(\frac{\xi}{\xi-z}\pm \frac{\bar{z}}{\bar{\xi}-\bar{z}}\right)\,d\theta.
\]

\item
Taking the plus sign in part (b), deduce the Poisson formula for the real part \(u(r,\phi)\)
of \(f(z)\) with \(z=re^{i\phi}\):
\[
u(r,\phi)=\frac{1}{2\pi}\int_0^{2\pi}
\frac{1-r^2}{1-2r\cos(\phi-\theta)+r^2}u(1,\theta)\,d\theta.
\]
This shows that the real part of \(f(z)\) on the unit circle determines the real part of \(f(z)\)
everywhere inside the unit circle.
\end{enumerate}

Topic Tags

Cauchy Integral Formula and Integral Theorems Harmonic Functions and Poisson Integral Formula Holomorphic Functions and Cauchy-Riemann Equations Real Integral Evaluation by Complex Methods