8. Let f(2) be defined for |z| < 1 by the series f(2) = E2,. Show that zf'(z) =- Log(1 – 2). Show that every analytic continuation of f satisfies e-"(=) = 1 – z. Showthat f can be analytically continued along every path that starts at 0, avoids 1, and neverreturns to 0.

We are given that z<1 and fz=∑zkk2k=1∞. We need to show that zf'z=-log1-z.

8. Let f(2) be defined for |2| < 1 by the series f(2) = E . Show that zf'(2) = - Log(1 – 2). Show that every analytic continuation of f satisfies e-j"(e) = 1 – z. Show that f can be analytically continued along every path that starts at 0, avoids 1, and never returns to 0.

8. Let f(2) be defined for |2| < 1 by the series f(2) = E . Show that zf'(2) = - Log(1 – 2). Show that every analytic continuation of f satisfies e-j"(e) = 1 – z. Show that f can be analytically continued along every path that starts at 0, avoids 1, and never returns to 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

8. Let f(2) be defined for |z| < 1 by the series f(2) = E2,. Show that zf'(z) =
- Log(1 – 2). Show that every analytic continuation of f satisfies e-"(=) = 1 – z. Show
that f can be analytically continued along every path that starts at 0, avoids 1, and never
returns to 0.

Expert Solution

Given

We are given that $|z| < 1$ and $f (z) = {\sum \frac{z^{k}}{k^{2}}}_{k = 1}^{\infty}$ .

We need to show that $z f' (z) = - \log (1 - z)$ .

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