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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