Poisson and Schwarz integral formulas

[10/100%]

Show

LaTeX

Show LaTeX

Problem 3.173 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Poisson's and Jensen's Formulas"

The function $f(z)$ is regular in the disk $\left|z\right|\leq R$ ; let $0<r<R$ . Then we have Poisson's formula

f(re^{i\vartheta }) = \frac{1}{2\pi }\int _{0}^{2\pi } f(Re^{i\Theta })\, \frac{R^{2}-r^{2}}{R^{2}-2Rr\cos (\Theta -\vartheta )+r^{2}}\, d\Theta .

Problem 3.174 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Poisson's and Jensen's Formulas"

Suppose that the function $f(z)$ is regular and bounded in the half-plane $\mathrm{Re}\left(z\right)\geq 0$ . Then, for $x>0$ ,

f(x+iy) = \frac{1}{\pi }\int _{-\infty }^{+\infty } f(i\eta )\, d\arctan \frac{\eta -y}{x}.

Problem 3.176 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Poisson's and Jensen's Formulas"

The function $f(z)$ is meromorphic in the disk $\left|z\right|\leq R$ , regular and different from $0$ on the circle $\left|z\right|=R$ and it has inside the circle the zeros $a_{1},a_{2},\ldots ,a_{m}$ and the poles $b_{1},b_{2},\ldots ,b_{n}$ , counted with correct multiplicities. If the point $z=re^{i\vartheta }$ , $r<R$ , is neither a zero nor a pole we have

\log \left|f(z)\right| + \sum _{\mu =1}^{m} \log \left|\frac{R^{2}-\overline{a_{\mu }}z}{(a_{\mu }-z)R}\right| - \sum _{\nu =1}^{n} \log \left|\frac{R^{2}-\overline{b_{\nu }}z}{(b_{\nu }-z)R}\right|

= \frac{1}{2\pi }\int _{0}^{2\pi } \log \left|f(Re^{i\Theta })\right|\, \frac{R^{2}-r^{2}}{R^{2}-2Rr\cos (\Theta -\vartheta )+r^{2}}\, d\Theta .

Problem 3.177 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Poisson's and Jensen's Formulas"

The function $f(z)$ is meromorphic in the half-plane $\mathrm{Re}\left(z\right)\geq 0$ , regular and non-zero on its boundary and it has inside the half-plane the zeros $a_{1},a_{2},\ldots ,a_{m}$ and the poles $b_{1},b_{2},\ldots ,b_{n}$ , counted with correct multiplicity. If $f(z)$ is regular at infinity (but also under weaker conditions on its behaviour at infinity) and if $f(z)$ is regular and non-zero at $z=x+iy$ , $x>0$ , we have

\log \left|f(z)\right| + \sum _{\mu =1}^{m} \log \left|\frac{z+\overline{a_{\mu }}}{z-a_{\mu }}\right| - \sum _{\nu =1}^{n} \log \left|\frac{z+\overline{b_{\nu }}}{z-b_{\nu }}\right|

= \frac{1}{\pi }\int _{-\infty }^{+\infty } \log \left|f(i\eta )\right|\, d\arctan \frac{\eta -y}{x}.

Problem 3.231 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "The Real Part of a Power Series"

(continued). We set $\re f(re^{i\vartheta }) = U(r,\vartheta )$ and assume that $f(0)$ is real and $\left|z\right|<r$ . Then

f(z) = \frac{1}{2\pi }\int _{0}^{2\pi } U(r,\vartheta )\, \frac{r+ze^{-i\vartheta }}{r-ze^{-i\vartheta }}\, d\vartheta .

Problem 3.232 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "The Real Part of a Power Series"

(continued). Suppose $f(z)$ does not vanish on $\left|z\right|=r$ and that its zeros in the disk $\left|z\right|<r$ are $c_{1},c_{2},\ldots ,c_{m}$ . Then we have for $\left|z\right|<r$

\log f(z) = i\gamma + \sum _{\mu =1}^{m} \log \frac{(z-c_{\mu })r}{r^{2}-\overline{c_{\mu }}z} + \frac{1}{2\pi }\int _{0}^{2\pi } \log \left|f(re^{i\vartheta })\right|\, \frac{r+ze^{-i\vartheta }}{r-ze^{-i\vartheta }}\, d\vartheta ,

where $\gamma$ is a real constant. [Consequence of problem 231 as problem 120 is a consequence of problem 119.]

Problem 31.1 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 31 "Application of Conformal Mappings to Solving Boundary Value Problems for Harmonic Functions"

Solve the classical Dirichlet problem for the unit disk $\left\{ z:\left|z\right| < 1\right\}$ :

\begin{cases} \Delta u = 0, & \left|z\right| < 1 ; \\ \left.u\right|_{\left|z\right| = 1} = \frac{1}{2+\sin \varphi }, & \text{ where } z = r e^{i \varphi } \end{cases}

Problem 31.2 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 31 "Application of Conformal Mappings to Solving Boundary Value Problems for Harmonic Functions"

Solve the general Dirichlet problem for the upper half-plane $\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}$ :

\begin{cases} \Delta u = 0, & y > 0 \\ \left.u\right|_{y = 0} = \sin x, & \text{ where } z = x+i y \end{cases}

Problem 31.3 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 31 "Application of Conformal Mappings to Solving Boundary Value Problems for Harmonic Functions"

Solve the general Dirichlet problem for the lower half-plane $\left\{ z: \mathrm{Im}\left(z\right) < 0\right\}$ :

\begin{cases} \Delta u = 0, & y < 0 \\ \left.u\right|_{\mathrm{Re}\left(z\right) \geqslant 0} = 1, & \left.u\right|_{\mathrm{Re}\left(z\right) < 0} = 0 \end{cases}

Problem 31.4 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 31 "Application of Conformal Mappings to Solving Boundary Value Problems for Harmonic Functions"

Solve the general Dirichlet problem for the exterior of the unit disk $\left\{ z:\left|z\right| > 1\right\}$ :

\begin{cases} \Delta u = 0, \quad \left|z\right| > 1 \\ \left.u\right|_{\left|z\right| = 1} = \frac{1}{5-4 \cos \varphi } \end{cases}

References

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.

[ШПК15] Шабунин М.И., Половинкин Е.С., Карлов М.И. Сборник задач по теории функций комплексного переменного. 4-е изд., БИНОМ. Лаборатория знаний, 2015.