91Ó°ÊÓ

\(f\) is called an odd function if \(f(z)=-f(-z)\) for all \(z ; f\) is called even if \(f(z)=f(-z)\). a. Show that an odd entire function has only odd terms in its power series expansion about \(z=0\). [Hint: show \(f\) odd \(\Rightarrow f^{\prime}\) even, etc., or use the identity $$ \left.f(z)=\frac{f(z)-f(-z)}{2} .\right] $$ b. Prove an analogous result for even functions.

By comparing the different expressions for the power series expansion of an entire function \(f\), prove that $$ f^{(k)}(0)=\frac{k !}{2 \pi i} \int_{C} \frac{f(\omega)}{\omega^{k+1}} d \omega, \quad k=0,1,2, \ldots $$ for any circle \(C\) surrounding the origin.

Prove that a nonconstant entire function cannot satisfy the two equations i. \(f(z+1)=f(z)\) ii. \(f(z+i)=f(z)\) for all \(z\). [Hint: Show that a function satisfying both equalities would be bounded.]

Suppose that \(f\) is entire and that for each \(z\), either \(|f(z)| \leq 1\) or \(\left|f^{\prime}(z)\right| \leq 1 .\) Prove that \(f\) is a linear polynomial. [Hint: Use a line integral to show $$ |f(z)| \leq A+|z| \text { where } A=\max (1,|f(0)|) .] $$

Find estimates for \(\sqrt{i}\) by applying Newton's method to the polynomial equation \(z^{2}=i\), with \(z_{0}=1 .\)