Exercises

Write a difference quotient for \(f''(w)\text{,}\) and use the formula for \(f'(w)\) in Theorem 5.1.1 to convert this difference quotient into an integral of \(f(z)\) divided by some polynomial.
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Subtract the desired integral formula for \(f''(w)\) from your integral for the difference quotient, and simplify to get the analogue of (5.1).
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Find a bound as in the proof of Theorem 5.1.1 for the integrand, and conclude that the limit of the difference quotient is the desired integral formula.
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3.

Integrate the following functions over the circle \(C[0,3]\text{:}\)

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(a)

\(\Log ( z - 4i )\)

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Answer.

\(0\)

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(b)

\(\ds \frac{ 1 }{ z - \frac 1 2 }\)

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Answer.

\(2 \pi i\)

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(c)

\(\ds \frac{ 1 }{ z^2 - 4 }\)

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Answer.

\(0\)

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(d)

\(\ds \frac{ \exp z }{ z^3 }\)

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Answer.

\(\pi i\)

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(e)

\(\ds \left( \frac{ \cos z }{ z } \right)^2\)

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Answer.

\(0\)

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(f)

\(i^{ z-3 }\)

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Answer.

\(0\)

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(g)

\(\ds \frac{ \sin z }{ (z^2 + \frac 1 2)^2 }\)

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(h)

\(\ds \frac{ 1 }{ (z+4) ( z^2 + 1 ) }\)

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(i)

\(\ds \frac{\exp(2z)}{(z-1)^2(z-2)}\)

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4.

Compute \(\ds \int_{ C[0,2] } \frac{ \exp z }{ (z-w)^2 } \, \diff{z}\) where \(w\) is any fixed complex number with \(|w| \ne 2\text{.}\)

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Answer.

\(2 \pi i \exp(w)\)

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5.

Define \(f : D[0,1] \to \C\) through

\begin{equation*} f(z) \ := \ \int_{ [0,1] } \frac{ \diff{w} }{ 1-wz } \end{equation*}

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(the integration path is from 0 to 1 along the real line). Prove that \(f\) is holomorphic in the unit disk \(D[0,1]\text{.}\)

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6.

To appreciate Corollary 5.1.6, show that the function \(f: \R \to \R\) given by

\begin{equation*} f(x) \ := \ \begin{cases}x^2 \sin(\frac 1 x) \amp \text{ if } x \ne 0 \, , \\ 0 \amp \text{ if } x = 0 \end{cases} \end{equation*}

is differentiable in \(\R\text{,}\) yet \(f'\) is not even continuous (much less differentiable) at 0.

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7.

Prove that \(f(z) = {\overline z}^2\) does not have an antiderivative in any nonempty region.

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8.

Show that \(\exp ( \sin z )\) has an antiderivative on \(\C\text{.}\) (What is it?)

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9.

Find a region on which \(f(z) = \exp( \frac 1 z)\) has an antiderivative. (Your region should be as large as you can make it. How does this compare with the real function \(f(x) = e^{\frac 1 x}\text{?}\))

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10.

Suppose \(f\) is continuous on \(\C\) and \(\lim_{z \to \infty} f(z)\) is finite. Show that \(f\) is bounded.

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Hint.

If \(\lim_{z\to \infty} f(z) = L\text{,}\) use the definition of the limit at infinity to show that there is \(R>0\) so that \(\abs{f(z)-L} \lt 1\) if \(\abs z >R\text{.}\)

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Now argue that \(|f(z)| \lt |L|+1\) for \(|z| > R\text{.}\) Use an argument from calculus to show that \(|f(z)|\) is bounded for \(\abs z\le R\text{.}\)

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11.

Let \(p\) be a polynomial of degree \(n>0\text{.}\) Prove that there exist complex numbers \(c, z_1, z_2, \dots, z_k\) and positive integers \(j_1, \dots , j_k\) such that

\begin{equation*} p(z) \ = \ c \left( z - z_1 \right)^{j_1} \left( z - z_2 \right)^{j_2} \cdots \left( z - z_k \right)^{j_k} \, \text{,} \end{equation*}

where \(j_1 + \dots + j_k = n\text{.}\)

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12.

Show that a polynomial of odd degree with real coefficients must have a real zero.

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Hint.

Use Exercise 1.5.24.

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13.

Suppose \(f\) is entire and \(|f(z)| \le \sqrt{ |z| }\) for all \(z \in \C\text{.}\) Prove that \(f\) is identically 0.

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Hint.

Show first that \(f\) is constant.

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14.

Suppose \(f\) is entire and there exists \(M > 0\) such that \(|f(z)| \ge M\) for all \(z \in \C\text{.}\) Prove that \(f\) is constant.

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15.

Suppose \(f\) is entire with bounded real part, i.e., writing \(f(z) = u(z) + i \, v(z)\text{,}\) there exists \(M > 0\) such that \(|u(z)| \le M\) for all \(z \in \C\text{.}\) Prove that \(f\) is constant.

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Hint.

Consider the function \(\exp(f(z))\text{.}\)

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16.

Suppose \(f\) is entire and there exist constants \(a\) and \(b\) such that \(|f(z)| \leq a|z| + b\) for all \(z \in \C\text{.}\) Prove that \(f\) is a polynomial of degree at most 1.

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Hint.

Use Theorem 5.1.1 and Exercise 2.5.30.

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17.

Suppose \(f: D[0,1] \to D[0,1]\) is holomorphic. Prove that for \(|z| \lt 1\text{,}\)

\begin{equation*} \left| f'(z) \right| \le \frac{ 1 }{ (1 - |z|)^2 } \,\text{.} \end{equation*}

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18.

Compute \(\ds \int_{ -\infty }^\infty \frac{ \diff{x} }{ x^4 + 1 } \text{.}\)

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Answer.

\(\frac{ \pi }{ \sqrt 2 }\)

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19.

In this problem \(f(z) = \frac{ \exp(iz) }{z^2+1}\) and \(R>1\text{.}\) Modify our computations in Example 5.3.5 as follows.

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(a)

Show that \(\int_{\sigma_R}f = \frac \pi e\) where \(\sigma_R\) is again (as in Figure 5.3.6) the counterclockwise semicircle formed by the segment \([-R,R]\) on the real axis, followed by the circular arc \(\gg_R\) of radius \(R\) in the upper half plane from \(R\) to \(-R\text{.}\)

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(b)

Show that \(\abs{\exp(iz)}\le1\) for \(z\) in the upper half plane, and conclude that \(\abs{f(z)}\le \frac 2 {\abs{z}^2}\) for sufficiently large \(|z|\text{.}\)

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(c)

Show that \(\lim_{R\to\infty}\int_{\gg_R} f = 0\) and hence \(\lim_{R\to\infty}\int_{[-R,R]} f = \frac \pi e\text{.}\)

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(d)

Conclude, by just considering the real part, that

\begin{equation*} \int_{-\infty}^\infty \frac{\cos(x)}{x^2+1}\,\diff{x} \ = \ \frac \pi e \, \text{.} \end{equation*}

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20.

Compute \(\ds \int_{-\infty}^\infty \frac{\cos(x)}{x^4+1}\,\diff{x} \text{.}\)

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Answer.

\(\displaystyle \frac{\pi}{e^{1/\sqrt{2}}} \sin\left(\frac{1}{\sqrt{2}} + \frac{\pi}{4}\right)\)

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21.

This exercise outlines how to extend some of the results of this chapter to the Riemann sphere as defined in Section 3.2. Suppose \(G \subseteq \C\) is a region that contains 0, let \(f\) be a continuous function on \(G\text{,}\) and let \(\gg \subset G \setminus \{0\}\) be a piecewise smooth path in \(G\) avoiding the origin, parametrized as \(\gg(t)\text{,}\) \(a \le t \le b\text{.}\)

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(a)

Show that

\begin{equation} \int_\gg f(z) \,\diff{z} \ = \ \int_\sigma f\!\left(\frac{1}{z}\right) \frac{1}{z^2} \, \diff{z} \tag{5.2} \end{equation}

where \(\sigma(t) := \frac{ 1 }{ \gg(t) }\text{,}\) \(a \le t \le b\text{.}\)

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(b)

Now suppose \(\gg\) is closed and \(\lim_{ z \to 0 } f\!\left(\frac{1}{z}\right) \frac{1}{z^2} = L\) is finite. Let \(H := \left\{ \frac 1 z : \, z \in G \setminus \{ 0 \} \right\}\) and define the function \(g: H \cup \{ 0 \} \to \C\) by

\begin{equation*} g(z) \ := \ \begin{cases}f\!\left(\frac{1}{z}\right) \frac{1}{z^2} \amp \text{ if } z \in H, \\ L \amp \text{ if } z=0. \end{cases} \end{equation*}

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Thus \(g\) is continuous on \(H \cup \{ 0 \}\) and (5.2) gives the identity

\begin{equation*} \int_\gg f \ = \ \int_\sigma g \,\text{.} \end{equation*}

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In particular, we can transfer certain properties between these two integrals. For example, if \(\int_\sigma g\) is path independent, so is \(\int_\gg f\text{.}\) Here is but one application:

Show that \(\int_\gg z^n \, \diff{z}\) is path independent for any integer \(n \ne -1\text{.}\)
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Conclude (once more) that \(\int_\gg z^n \, \diff{z} = 0\) for any integer \(n \ne -1\text{.}\)
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