Chapter 6

Recall that \(C_{\rho}^{+}\left(z_{0}\right)\) denotes the positively oriented circle \(\left\\{z:\left|z-z_{0}\right|=\rho\right\\}\). Find \(\int_{C_{1}^{+}(0)}(\exp z+\cos z) z^{-1} d z\)

Factor each polynomial as a product of linear factors. (a) \(P(z)=a^{4}+4\) (b) \(P(z)=z^{2}+(1+i) z+5 i\) (c) \(P(z)=x^{4}-4 z^{3}+6 z^{2}-4 z+5\) (d) \(P(z)=z^{3}-(3+3 i) z^{2}+(-1+6 i) z+3-i\). Hint: Show that \(P(i)=0\).

Let \(m\) and \(n\) be integers. Show that \(\int_{0}^{2 \pi} e^{i m t} e^{-i n t} d t=\left\\{\begin{array}{l}0 \text { when } m \neq n \\ 2 \pi \text { when } m=n\end{array}\right.\)

Sketch the following curves. (a) \(z(t)=t^{2}-1+i(t+4)\), for \(1 \leq t \leq 3\) (b) \(z(t)=\sin t+i \cos 2 t\), for \(-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}\). (c) \(z(t)=5 \cos t-i 3 \sin t\), for \(\frac{\pi}{2} \leq t \leq 2 \pi\).

Show that \(\int_{C} z^{-1} d z=2 \pi i\), where \(C\) is the square with vertices \(1 \pm i\) and \(-1 \pm i\) and having positive orientation.

Recall that \(C_{\rho}^{+}\left(z_{0}\right)\) denotes the positively oriented circle \(\left\\{z:\left|z-z_{0}\right|=\rho\right\\}\). Find \(\int_{C_{1}^{+}(1)}(z+1)^{-1}(z-1)^{-1} d z\)

Show that \(\int_{0}^{\infty} e^{-z t} d t=\frac{1}{4}\) provided \(\operatorname{Re}(z)>0 .\)

Consider the integral \(\int_{C} z^{2} d z\), where \(C\) is the positively oriented upper semicircle of radius 1 , centered at 0 . (a) Give a Riemann sum approximation for the integral by selecting \(n=4\) and the points \(z_{k}=e^{i \frac{k}{4}}(k=0, \ldots, 4)\) and \(c_{k}=e^{i \frac{(2 k-1) \tau}{8}}(k=1, \ldots, 4)\). (b) Compute the integral exactly by selecting a parametrization for \(C\) and applying Theorem 6.1.

Show that \(\int_{C_{1}^{+}(0)}\left(4 z^{2}-4 z+5\right)^{-1} d z=0\).

Recall that \(C_{\rho}^{+}\left(z_{0}\right)\) denotes the positively oriented circle \(\left\\{z:\left|z-z_{0}\right|=\rho\right\\}\). Find \(\int_{C_{1}^{+}(1)}(z+1)^{-1}(z-1)^{-2} d z\)

Show that \(\cos z\) is not a bounded function.

Find \(\int_{C}\left(z^{2}-z\right)^{-1} d z\) for (a) circle \(C=C_{2}^{+}(1)=\\{z:|z-1|=2\\}\) having positive orientation. (b) circle \(C=C_{j}^{+}(1)=\left\\{z:|z-1|=\frac{1}{2}\right\\}\) having positive orientation.

Recall that \(C_{\rho}^{+}\left(z_{0}\right)\) denotes the positively oriented circle \(\left\\{z:\left|z-z_{0}\right|=\rho\right\\}\). Find \(\int_{C_{1}^{+}(1)}\left(z^{3}-1\right)^{-1} d z\)

Let \(f(z)=z^{2}\). Evaluate the following, where \(R\) represents the rectangular region defined by the set \(R=\\{z=x+i y: 2 \leq x \leq 3\) and \(1 \leq y \leq 3\\}\). (a) \(\max _{x \in R}|f(z)|\). (b) \(\left.\min _{x \in R} \mid f(z)\right]\). (c) \(\max _{x \in R} \operatorname{Re}[f(z)]\). (d) \(\left.\min _{z \in R} \operatorname{lm} \mid f(z)\right]\).

Let \(f(t)=u(t)+i v(t)\), where \(u\) and \(v\) are differentiable. Show that \(\int_{a}^{b} f(t) f^{\prime}(t) d t=\frac{1}{2}[f(b)]^{2}-\frac{1}{2}[f(a)]^{2} .\)

Find \(\int_{c}(2 z-1)\left(z^{2}-z\right)^{-1} d z\) for the (a) circle \(C=C_{2}^{+}(0)=\\{z:|z|=2\\}\) having positive orientation. (b) circle \(C=C_{j}^{+}(0)=\left\\{z:|z|=\frac{1}{2}\right\\}\) having positive orientation.

Recall that \(C_{\rho}^{+}\left(z_{0}\right)\) denotes the positively oriented circle \(\left\\{z:\left|z-z_{0}\right|=\rho\right\\}\). Find \(\int_{C_{1}^{+}(0)} z^{-4} \sin z d z\)

Recall that \(C_{\rho}^{+}\left(z_{0}\right)\) denotes the positively oriented circle \(\left\\{z:\left|z-z_{0}\right|=\rho\right\\}\). Find \(\int_{C_{1}^{+}(0)}(z \cos z)^{-1} d z\)

Let \(f\) be an entire function such that \(|f(z)| \leq M|z|\) for all \(z\). (a) Show that, for \(n \geq 2, f^{(n)}(z)=0\) for all \(z\). (b) Use part (a) to show that \(f(z)=a z+b\).

Recall \(C_{r}^{+}(a)\) is the circle of radius \(r\) centered at \(a\), oriented eounterclockwise. (a) Evaluate \(\int_{C_{i}^{+}(0)} z d z\). (b) Evaluate \(\int_{\sigma_{4}^{+}(0)} \bar{y} d z\). (c) Evaluate \(\int_{C_{2}^{-}(0)} \frac{1}{x} d z\). (The minus sign means elockwise orientation.) (d) Evaluate \(\int_{C_{2}^{-}(0)} \frac{1}{2} d z\). (e) Evaluate \(\int_{C}(z+1) d z\), where \(C\) is \(C_{1}^{+}(0)\) in the first quadrant. (f) Evaluate \(\int_{c}\left(x^{2}-i y^{2}\right) d z\), where \(C\) is the upper half of \(C_{1}^{+}(0)\). (g) Evaluate \(\int_{C}|z-1|^{2} d z\), where \(C\) is the upper half of \(C_{1}^{+}(0)\).

Evaluate \(\int_{C}\left(4 z^{2}+4 z-3\right)^{-1} d z=\int_{C}(2 z-1)^{-1}(2 z+3)^{-1} d z\) for (a) the circle \(C=C_{1}^{+}(0)\). (b) the circle \(C=C_{1}^{+}\left(-\frac{2}{3}\right)=\left\\{z:\left|z+\frac{2}{3}\right|=1\right\\}\). (c) the circle \(C=C_{3}^{+}(0)\).

Establish the following minimum modulus principle. (a) Let \(f\) be analytie and nonconstant in the domain \(D .\) If \(|f(z)| \geq m\) for all \(z\) in \(D\), where \(m>0\), then \(|f(z)|\) does not attain a minimum value at any point \(z_{0}\) in \(D\). (b) Show that the requirement \(m>0\) in part a is necessary by finding an example of a function defined on \(D\) for which \(m=0\), and yet whose minimum is attained somewhere in \(D\).

Use Green's theorem to show that the area enclosed by a simple closed contour \(C\) is \(\frac{1}{2} \int_{C} x d y-y d x\).

Let \(u(x, y)\) be harmonic for all \((x, y)\). Show that $$ u\left(x_{0}, y_{0}\right)=\frac{1}{2 \pi} \int_{0}^{2 \pi} u\left(x_{0}+R \cos \theta, y_{0}+R \sin \theta\right) d \theta $$ where \(R>0 .\) Hint: Let \(f(z)=u(x, y)+i v(x, y)\), where \(v\) is a harmonic conjugate of \(u\).

Establish the following maximum principle for harmonic functions. Let \(u(x, y)\) be harmonic and nonconstant in the simply connected domain \(D\). Then u does not have a maximum value at any point \(\left(x_{0}, y_{0}\right)\) in \(D\).

Find \(\int_{C} z^{-2}\left(z^{2}-16\right)^{-1} \exp z d z\) along the following contours: (a) The circle \(C_{1}^{+}(0)\). (b) The circle \(C_{1}^{+}(4)\).

Let \(f\) be an entire function with the property that \(|f(z)| \geq 1\) for all \(z\). Show that \(f\) is constant.

Evaluate \(\int_{C} z^{2} d z\), where \(C\) is the line segment from 1 to \(1+i\).

Let \(f\) be a nonconstant analytic function in the closed disk \(\bar{D}_{1}(0)\). Suppose that \(|f(z)|=K\) for \(z \in C_{1}(0)\). Show that \(f\) has a zero in \(D\). Hint: Use both the maximum and minimum modulus principles.

Evaluate \(\int_{C}\left|z^{2}\right| d z\), where \(C\) is given by \(C: z(t)=t+i t^{2}\), for \(0 \leq t \leq 1\).

Suppose that \(f(z)=u(r, \theta)+i v(r, \theta)\) is analytic for all values of \(z=r e^{i \theta}\). Show that \(\int_{0}^{2 \pi}[u(r, \theta) \cos \theta-v(r, \theta) \sin \theta] d \theta=0\) Hint: Integrate \(f\) around the circle \(C_{1}^{+}(0)\).

Find \(\int_{C} z^{-1}(z-1)^{-1} \exp z d z\) along the following contours: (a) The circle \(C_{j}^{+}(0)\). (b) The circle \(C_{2}^{+}(0)\).

Why is it important to study the fundamental theorem of algebra in a complex analysis course?

Evaluate \(\int_{C} \exp z \mathrm{~d} s\), where \(C\) is the straight-line segment joining 1 to \(1+i \pi\).

Evaluate \(\int_{C}^{z} \exp z d z\), where \(C\) is the square with vertices \(0,1,1+i\), and \(i\) taken with the counterclock wise orientation.

Find \(\int_{C_{3}^{+}(1)}\left(z^{2}+1\right)^{-2} d z\).

Evaluate \(\int_{C} \exp z d z\), where \(C\) is the straight-line segment joining 0 to \(1+i\).

Show that \(\int_{C} 1 d z=z_{2}-z_{1}\), where \(C\) is the line segment from \(z_{1}\) to \(z_{2}\), by parametrizing \(C .\)

Let \(z(t)=x(t)+i y(t)\), for \(a \leq t \leq b\), be a smooth curve. Give a meaning for each of the following expressions. (a) \(z^{\prime}(t)\). (b) \(\left|z^{\prime}(t)\right| d t\). (c) \(\int_{a}^{b} z^{\prime}(t) d t\) (d) \(\int_{a}^{b}\left|z^{\prime}(t)\right| d t\)

Let \(P(z)=a_{0}+a_{1} z+a_{2} z^{2}+a_{3} z^{3}\). Find \(\int_{C_{1}^{+}(0)} P(z) z^{-n} d z\), where \(n\) is a positive integer.

Evaluate \(\int_{0} \cos z d z\), where \(C\) is the polygonal path from 0 to \(1+i\) that consists of the line segments from 0 to 1 and 1 to \(1+i\).

Let \(I_{1}\) and \(z_{2}\) be two complex numbers that lie interior to the simple closed contour \(C\) with positive orientation. Evaluate \(\int_{C}\left(z-z_{1}\right)^{-1}\left(z-z_{2}\right)^{-1} d z\).

Let \(f(t)=e^{i t}\) be defined on \(a \leq t \leq b\), where \(a=0\), and \(b=2 \pi .\) Show that there is no number \(c \in(a, b)\) such that \(f(c)(b-a)=\int_{0}^{b} f(t) d t .\) In other words, the mean value theorem for definite integrals that you learned in ealculus does not hold for complex funetions.

Using partial fraction decomposition, show that if \(z\) lies in the right half- plane and \(C\) is the line segment joining 0 to \(z\), then $$ \int_{C} \frac{d \xi}{\xi^{2}+1}=\operatorname{Arctan} z=\frac{i}{2} \log (z+i)-\frac{i}{2} \log (z-i)+\frac{\pi}{2} $$

Let \(f\) be analytic in the simply connected domain \(D\) and let \(z_{1}\) and \(z_{2}\) be two complex numbers that lie interior to the simple closed contour \(C\) having positive orientation that lies in \(D\). Show that, \(\frac{f\left(z_{2}\right)-f\left(z_{1}\right)}{z_{2}-z_{1}}=\frac{1}{2 \pi i} \int_{C} \frac{f(z) d z}{\left(z-z_{1}\right)\left(z-z_{2}\right)}\) State what happens when \(z_{2} \rightarrow z_{1}\).

Use the ML inequality to show that \(\left|P_{n}(x)\right| \leq 1\), where \(P_{n}\) is the nth Legendre polynomial defined on \(-1 \leq x \leq 1\) by \(P_{n}(x)=\frac{1}{x} \int_{0}^{\pi}\left(x+i \sqrt{1-x^{2}} \cos \theta\right)^{n} d \theta\).

Let \(f^{\prime}\) and \(g^{\prime}\) be analytic for all \(z\) and let \(C\) be any contour joining the points \(z_{1}\) and \(z_{2}\). Show that $$ \int_{C} f(z) g^{\prime}(z) d z=f\left(z_{2}\right) g\left(z_{2}\right)-f\left(z_{1}\right) g\left(z_{1}\right)-\int_{C} f^{\prime}(z) g(z) d z $$

The Legendre polynomial \(P_{n}(z)\) is defined by \(P_{n}(z)=\frac{1}{2^{n} n !} \frac{d^{n}}{d z^{n}}\left[\left(z^{2}-1\right)^{n}\right]\) Use Cauchy's integral formula to show that \(P_{n}(z)=\frac{1}{2 \pi i} \int_{C} \frac{\left(\xi^{2}-1\right)^{n} d \xi}{2^{n}(\xi-z)^{n+1}}\) where \(C\) is a simple closed contour having positive orientation and \(z\) lies inside \(C\).

Explain how contour integrals in complex analysis and line integrals in calculus are different. How are they similar?