# AMAT 415 — Weeks 6 & 7 problems

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28) Find all the solutions of $\cosh(z) = i$.  (Hint: Rewrite the equations in terms of exponentials and manipulate it into a quadratic equation in the variable $e^z$.)

29) Suppose $f(z)$ is analytic on $\mathbb{C}$.  Let $g(z)=f(\bar{z})$.  For which $z$ is $g(z)$ analytic? (Hint: Use the Cauchy-Riemann equations.)

30) Suppose $f(z)$ is analytic on $\mathbb{C}$.  Let $h(z)=\overline{f(\bar{z})}$.  For which $z$ is $h(z)$ analytic?

31) Find a function $v(x,y)$ such that $v(0,0)=2$ and $f(z)=x^3-3xy^2+iv(x,y)$ is analytic.

32) Sketch and describe in a few words the paths $\gamma(t)=1+\cos(t) + i\sin(t)$, $t\in [\pi/2,\pi]$, and $\gamma(t)=2\cos(t) + i\sin(t)$, $t\in [0,\pi]$.

33) Evaluate $\int_\gamma xdz$, where $\gamma$ is the union of the line segments joining $0$ to $i$ and then $i$ to $i+2$.

34) Evaluate $\int_\gamma e^z dz$, where $\gamma$ is the part of the unit circle joining $1$ to $i$ in a counterclockwise direction.

35) Evaluate $\int_\gamma \frac{1}{|z|}dz$, where $\gamma$ is the unit circle traversed counterclockwise.

36) In class, we showed that $\int_\gamma \frac{1}{z}dz=2\pi i$, where $\gamma(t)=e^{2\pi i t}$, $t\in [0,1]$.  Identify the flaw in the following argument:  Since $\log(z)$ is an antiderivative of $1/z$, the fundamental theorem of calculus implies that $\int_\gamma \frac{1}{z}dz=\log(\gamma(1))-\log(\gamma(0))=1-1=0$.

37) Evaluate $\int_\gamma \frac{1}{z-i}dz$ where $\gamma$ is the circle centered at $z=0$ with radius $3$. (Hint: Deform the path $\gamma$ into a more convenient one.)

38) Evaluate $\int_\gamma\frac{1}{(z-1)^3}dz$, where $\gamma$ is the (a) circle or radius $1/2$ centered at $z=0$; (b) the circle of radius $1/2$ centered at $z=1$; (c) the circle of radius $z=1/2$ centered at $z=-1$.

39) Let

$\displaystyle{f(x)=\begin{cases} x^2\sin(\frac{1}{x})&x\neq 0,\\ 0&x=0\end{cases}}$.

Show that

$\displaystyle{f'(x)=\begin{cases} 2x\sin(\frac{1}{x})-\cos(\frac{1}{x})&x\neq 0,\\ 0&x=0\end{cases}}$.

Conclude that $f'$ is not continuous at $x=0$. (Hint: Use the definition of the derivative to compute $f'(0)$.) Challenge: Adapt this exercise to construct a function such that $f'(x)$, $f''(x)$, $\ldots$, $f^{(n)}(x)$ exist but such that $f^{(n)}(x)$ is discontinuous at $x=0$.

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