Chapter 7

By computing derivatives, find the Maclaurin series for each function and state where it is valid. (a) \(\sinh z\), (b) \(\cosh z\). (c) \(\log (1+z)\).

Locate the zeros of the following functions and determine their order. (a) \(\left(1+z^{2}\right)^{4}\). (b) \(\sin ^{2} z\). (c) \(z^{2}+2 z+2\) (d) \(\sin z^{2}\). (e) \(z^{4}+10 z^{2}+0\) (f) \(1+\exp z\). (g) \(z^{6}+1\). (h) \(z^{3} \exp (z-1)\). (i) \(z^{6}+2 z^{3}+1\). (j) \(z^{3} \cos ^{2} z\). (k) \(z^{n}+z^{4}\). (1) \(z^{2} \cosh z\).

Determine whether there exists a function \(f\) that is analytic at 0 such that for \(n=1,2,3, \ldots\), (a) \(f\left(\frac{1}{2 n}\right)=0\) and \(f\left(\frac{1}{2 n-1}\right)=1\). (b) \(f\left(\frac{1}{n}\right)=f\left(\frac{-1}{n}\right)=\frac{1}{n}\). (c) \(f\left(\frac{1}{n}\right)=f\left(\frac{-1}{n}\right)=\frac{1}{n}\).

Locate the poles of the following functions and determine their order. (a) \(\left(z^{2}+1\right)^{-3}(z-1)^{-4}\). (b) \(z^{-1}\left(z^{2}-2 z+2\right)^{-2}\). (c) \(\left(z^{6}+1\right)^{-1}\). (d) \(\left(z^{4}+z^{3}-2 z^{2}\right)^{-1}\). (e) \(\left(3 z^{4}+10 z^{2}+3\right)^{-1}\). (f) \(\left(i+\frac{2}{x}\right)^{-1}\left(3+\frac{4}{x}\right)^{-1}\). (g) \(z \cot z\). (h) \(z^{-5} \sin z\) (i) \(\left(z^{2} \sin z\right)^{-1}\). (j) \(={ }^{-1} \csc z\). \((k)(1-\exp z)^{-1}\) (1) \(z^{-5} \sinh z\).

Find the Taylor series centered at \(a=1\) and state where it converges for (a) \(f(z)=\frac{1-x}{x-2}\). (b) \(f(z)=\frac{1-1}{x-3}\). Hint: \(\frac{1-s}{x-1}=\left(\frac{1}{2}\right) \frac{x-1}{1-\frac{1-1}{2}}=\left(\frac{1}{2}\right)(z-1) \frac{1}{1-\frac{s-1}{2}}\).

Show that \(S_{n}(z)=\sum_{h=0} z^{k}=\frac{1-z^{n}}{1-x}\) does not converge uniformly to \(f(z)=\frac{1}{1-x}\) on the set \(T=D_{1}(0)\) by appealing to Statement \((7-3)\). Hint: Given \(\varepsilon>0\) and \(\mathrm{a}\) positive integer \(n\), let \(z_{n}=\varepsilon^{\frac{1}{n}}\).

Let \(f\) be analytic and have a zero of order \(k\) at zo. Show that \(f^{\prime}\) has a zero of order \(k-1\) at \(z_{0}\) -

Show that the real function \(f\) defined by \(f(x)=\left\\{\begin{array}{l}x \sin \left(\frac{1}{x}\right) \text { when } x \neq 0, \text { and } \\ 0 & \text { when } x=0\end{array}\right.\) is continuous at \(x=0\) but that the corresponding function \(g(z)\) defined by \(g(z)=\left\\{\begin{array}{l}z \sin \left(\frac{1}{z}\right) \text { when } z \neq 0, \text { and } \\ 0 \quad \text { when } z=0\end{array}\right.\) is not continuous at \(z=0\).

Let \(f(z)=(1+z)^{\beta}=\exp [\beta \log (1+z)]\) be the principal branch of \((1+z)^{\beta}\), where \(\beta\) is a fixed complex number. Establish the validity for \(z \in D_{1}(0)\) of the binomial expanaion \(\begin{aligned}(1+z)^{\theta} &=1+\beta z+\frac{\beta(\beta-1)}{2 !} z^{2}+\frac{\beta(\beta-1)(\beta-2)}{3 !} z^{3}+\cdots \\\ &=1+\sum_{n=1}^{\infty} \frac{\beta(\beta-1)(\beta-2) \cdots(\beta-n+1)}{n !} z^{n} . \end{aligned}\)

Suppose that the sequences of functions \(\left\\{f_{n}\right\\}\) and \(\left\\{g_{n}\right\\}\) converge uniformly on the set \(T\). (a) Show that the sequence \(\left\\{f_{n}+g_{n}\right\\}\) converges uniformly on the set \(T\). (b) Show by example that it, is not necessarily the case that \(\left\\{f_{n} g_{n}\right\\}\) converges uniformly on the set \(T\).

Let \(f\) and \(g\) have poles of order \(m\) and \(n\), reapectively, at \(z_{0}\). Show that \(f+g\) has either a pole or a removable singularity at \(z_{0}\)

Suppose that \(f(z)=\sum_{n=0} c_{n} z^{n}\) is an entire function. (a) Find a series representation for \(\overline{f(\bar{z})}\), using powers of \(\mathrm{z}\). (b) Show that \(\overline{f(z)}\) is an entire function. (c) Does \(\overline{f(z)}=f(z) ?\) Why or why not?

Find the Laurent series for \(f(z)=\frac{1}{x^{4}(1-z)^{2}}\) that involves powers of \(z\) and is valid for \(|z|>1\). Hint: \(\frac{1}{\left(1-\frac{1}{2}\right)^{2}}=\frac{a^{a}}{(1-z)^{2}}\).

Let \(f\) be analytic and have a zero of order \(k\) at \(z_{0} .\) Show that the funetion \(\frac{1}{f}\) has a simple pole at zo.

Consider the function \(\zeta(z)=\sum_{n=1}^{\infty} n^{-x}\), where \(n^{-t}=\exp (-z \ln n)\) (a) Show that \(\zeta(z)\) converges uniformly on the set \(A=\\{z: \operatorname{Re}(z) \geq 2\\}\). (b) Let \(D\) be a closed disk contained in \(\\{z: \operatorname{Re}(z)>1\\}\). Show that \(\zeta(z)\) converges uniformly on \(D\).

Let \(f\) have a pole of order \(k\) at \(\Sigma_{0}\). Show that \(f^{\prime}\) has a pole of order \(k+1\) at \(z_{0}\).

Find the singularities of the following functions. (a) \(\frac{1}{\sin (1 / x)}\). (b) \(\log z^{2}\). (c) \(\cot z-\frac{1}{x}\)

Can Log z be represented by a Maclaurin series or a Laurent series about the point \(\alpha=0 ?\) Explain your answer.

How are the definitions of singularity in complex analysis and asymptote in calculus different? How are they similar?

Use the Maclaurin series for \(\sin z\) and then long division to get the Laurent series for \(\csc z\) with \(\alpha=0\).

Compute the Taylor series for the principal logarithm \(f(z)=\log z\) expanded about the center \(z_{0}=-1+i\).

Let \(f\) be defined in a domain that contains the origin. The function \(f\) is said to be even if \(f(-z)=f(z)\), and it is called odd if \(f(-z)=-f(z)\). (a) Show that the derivative of an odd function is an even function. (b) Show that the derivative of an even function is an odd function. Hint: Use limits. (c) If \(f(z)\) is even, show that all the coefficients of the odd powers of \(z\) in the Maclaurin series are zero. (d) If \(f(z)\) is odd, show that all the coefficients of the even powers of \(z\) in the Maclaurin series are zero.

The Z-transform. Let \(\left\\{a_{n}\right\\}\) be a sequence of complex numbers satisfying the growth condition \(\left|a_{n}\right| \leq M R^{n}\) for \(n=0,1, \ldots\) and for some fixed positive values \(M\) and \(R\). Then the \(Z\) -transform of the sequence \(\left\\{a_{n}\right\\}\) is the function \(F(z)\) defined by \(Z\left(\left\\{a_{n}\right\\}\right)=F(z)=\sum_{n=0}^{\infty} a_{n} z^{-n}\) (a) Prove that \(F(z)\) converges for \(|z|>R\). (b) Find \(Z\left(\left\\{a_{n}\right\\}\right)\) for i. \(a_{n}=2\). ii. \(a_{n}=\frac{1}{n}\). iii. \(a_{\mathrm{n}}=\frac{1}{n+1}\). iv. \(a_{n}=1\), when \(n\) is even, and \(a_{n}=0\) when \(n\) is odd. (c) Prove that \(Z\left(\left\\{a_{n+1}\right\\}\right)=z\left[Z\left(\left\\{a_{n}\right\\}\right)-a_{0}\right] .\) This relation is known as the shifting property for the \(Z\) -transform.

Consider the real-valued function \(f\) defined on the real numbers as \(f(x)=\left\\{\begin{array}{c}e^{-\frac{1}{x^{2}}} \text { when } x \neq 0 \\\ 0 \quad \text { when } x=0\end{array}\right.\) (a) Show that, for all \(n>0, f^{(n)}(0)=0\), where \(f^{(n)}\) is the \(n\) th derivative of \(f\). Hint: Use the limit definition for the derivative to establish the case for \(n=1\) and then use mathematical induction to complete your argument. (b) Explain why the function \(f\) gives an example of a function that, although differentiable everywhere on the real line, is not expressible as a. Taylor series about 0. Hint: Evaluate the Taylor series representation for \(f(x)\) when \(x \neq 0\), and show that the series does not equal \(f(x)\). (c) Explain why a similar argument could not be made for the complexvalued function \(g\) defined on the complex numbers as \(g(z)=\left\\{\begin{array}{c}e^{-\frac{1}{x^{2}}} \text { when } z \neq 0 \\\ 0 \quad \text { when } z=0\end{array}\right.\) Hint: Show that \(g(z)\) is not even continuous at \(z=0\) by taking limits along the real and imaginary axes.