Coefficient extraction and contour-deformation tricks

[2/100%]

Show

LaTeX

Show LaTeX

Problem 3.159 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Cauchy's Formula"

(continued). We find

\frac{1}{2\pi i}\int _{L} e^{e^{\zeta }}\, d\zeta = 1.

Problem 3.27 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 3 "Limit and Continuity of a Function of a Complex Variable. Integration of a Function of a Complex Variable"

Let the function $f(z)$ be continuous in the angle

-\alpha \leqslant \arg z \leqslant \alpha \quad (0 < \alpha < \pi ),

and let $z f(z) \rightarrow A$ as $z \rightarrow \infty , \quad \left|\arg z\right| \leqslant \alpha$ . Denote by $C_{R}$ the arc of the circle $\left|z\right| = R, \quad \left|\arg z\right| \leqslant \alpha$ , going from the point $z = R e^{-i \alpha }$ to the point $z = R e^{i \alpha }$ . Prove that

\lim _{R \rightarrow \infty } \int _{C_{R}} f(z) d z = 2 i \alpha A

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.