Problem 17
Show that \(\left|e^{-z}\right|<1\) if and only if \(\operatorname{Re}(z)>0\).
Problem 18
Show that each of the following functions - call each one \(u(x, y)-\) is harmonic, and find the function's harmonic partner, \(v(x, y)\), such that \(u(x, y)+i v(x, y)\) is analytic. (a) \(x^{3}-3 x y^{2}\); (b) \(e^{x} \cos y\); (c) \(\frac{x}{x^{2}+y^{2}}, \quad\) where \(x^{2}+y^{2} \neq 0\); (d) \(e^{-2 y} \cos 2 x\); (e) \(e^{y^{2}-x^{2}} \cos 2 x y\); (f) \(e^{x}(x \cos y-y \sin y)+2 \sinh y \sin x+x^{3}-3 x y^{2}+y\).
Problem 19
Prove the following identities. (a) \(\cos ^{-1} z=-i \ln \left(z \pm \sqrt{z^{2}-1}\right)\), (b) \(\sin ^{-1} z=-i \ln \left[i z \pm \sqrt{1-z^{2}}\right]\), (c) \(\tan ^{-1} z=\frac{1}{2 i} \ln \left(\frac{i-z}{i+z}\right)\), (d) \(\quad \cosh ^{-1} z=\ln \left(z \pm \sqrt{z^{2}-1}\right)\), (e) \(\sinh ^{-1} z=\ln \left(z \pm \sqrt{z^{2}+1}\right)\), (f) \(\tanh ^{-1} z=\frac{1}{2} \ln \left(\frac{1+z}{1-z}\right)\).
Problem 20
Find the curve defined by each of the following equations.
(a) \(z=1-i t, \quad 0 \leq t \leq 2\),
(b) \(z=t+i t^{2}, \quad-\infty
Problem 22
Let \(f(t)=u(t)+i v(t)\) be a (piecewise) continuous complex-valued function of a real variable \(t\) defined in the interval \(a \leq t \leq b\). Show that if \(F(t)=U(t)+i V(t)\) is a function such that \(d F / d t=f(t)\), then $$\int_{a}^{b} f(t) d t=F(b)-F(a) .$$ This is the fundamental theorem of calculus for complex variables.
Problem 23
Find the value of the integral \(\int_{C}[(z+2) / z] d z\), where \(C\) is (a) the semicircle \(z=2 e^{i \theta}\), for \(0 \leq \theta \leq \pi\), (b) the semicircle \(z=2 e^{i \theta}\), for \(\pi \leq \theta \leq 2 \pi\), and (c) the circle \(z=2 e^{i \theta}\), for \(-\pi \leq \theta \leq \pi\).
Problem 24
Evaluate the integral \(\int_{\gamma} d z /(z-1-i)\) where \(\gamma\) is (a) the line joining \(z_{1}=2 i\) and \(z_{2}=3\), and (b) the broken path from \(z_{1}\) to the origin and from there to \(z_{2}\).
Problem 25
Evaluate the integral \(\int_{C} z^{m}\left(z^{*}\right)^{n} d z\), where \(m\) and \(n\) are integers and \(C\) is the circle \(|z|=1\) taken counterclockwise.
Problem 26
Let \(C\) be the boundary of the square with vertices at the points \(z=0\), \(z=1, z=1+i\), and \(z=i\) with counterclockwise direction. Evaluate $$\oint_{C}(5 z+2) d z \text { and } \oint_{C} e^{\pi z^{*}} d z$$
Problem 27
Use the definition of an integral as the limit of a sum and the fact that absolute value of a sum is less than or equal to the sum of absolute values to prove the Darboux inequality.
