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Show that \(\int_{|z|=R}|(\sin z) / z||d z| \rightarrow \infty\) as $R \rightarrow \infty$.

Short Answer

Expert verified
By evaluating the integral, applying the Estimation Lemma and analyzing the resulting expression, it is evident that as \(R\) approaches infinity, the integral \(\int_{|z|=R}|\frac{\sin z}{z}||dz|\) indeed tends to infinity.
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Step 1: Represent sin(x) for Complex Numbers

The power series representation is used to rewrite the \(\sin(z)\) for complex numbers. It can be represented as: \(\sin(z) = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \dots + \frac{(-1)^n(z^{2n+1})}{(2n+1)!} + \dots\) This is important to accomplish the integration in the next steps.

Step 2: Substitution and Evaluation of the Integral

Now substitute \(\sin(z)\) from Step 1 into the integral and then divide by \(z\). The resulting expression is: \(\int_{|z|=R} |1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \frac{z^6}{7!} + \dots + \frac{(-1)^n(z^{2n})}{(2n+1)!} + \dots||dz|\). Following, generate an expression that provides an upper estimate of the integral by making every term in absolute value.

Step 3: Implementation of Estimation Lemma

Estimation Lemma comes handy when upper bound of a function on a circle of radius \(R\) is needed to be found: Estimation Lemma states that \(\left|\int_{c}f(z)dz\right| \leq ML\) where \(M\) is maximum of \(|f(z)|\) on \(c\) and \(L\) is the length of the curve \(c\). Here, \(L = 2 \pi R\) as the length of the contour \(|z|=R\) is the circumference of the circle with radius \(R\). This is used on the expression in Step 2 to find its upper bound.

Step 4: Analyzing the Resulting Expression

The upper estimation yields \(2 \pi R \cdot M\) as \(R\to\infty\). As \(R\) is a factor in the result and approaches to infinity, \(2 \pi R \cdot M\) will also approach infinity, even if \(M\) does not dependent on \(R\) as long as it is non-zero. Thus, it is indeed shown that \(\int_{|z|=R}|\frac{\sin z}{z}||dz| \rightarrow \infty\), when \(R \rightarrow \infty\).

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Most popular questions from this chapter

Chapter 9

Determine the residue at each singularity for the following functions. (a) \(\frac{1}{\cos z}\) (b) \(\frac{z}{(z-1)^{2}(z-2)}\) (c) \(z^{n} \cos \frac{1}{z}\)

Chapter 9

Suppose \(f(z)\) is analytic at \(z_{0}\) with $f^{\prime}\left(z_{0}\right) \neq 0 .\( Show that there exists an analytic function \)g(z)\( such that \)f(g(z))=z$ in some neighborhood of \(z_{0}\). This is known as the inverse function theorem.

Chapter 9

Find the principal part for the following Laurent series. (b) \(\frac{z^{2}}{z^{4}-1} \quad(0<|z+i|<\sqrt{2})\) (c) \(\frac{e^{z}}{z^{4}} \quad(|z|>0)\) (d) \(\frac{\sin z}{z^{4}} \quad(|z|>0)\) (e) \(\frac{1}{\tan ^{2} z}-\frac{1}{z^{2}} \quad(0<|z|<\pi / 2)\).

Chapter 9

Use contour integration to evaluate (a) \(\int_{0}^{\infty} \frac{d x}{1+x^{6}}\) (b) \(\int_{0}^{\infty} \frac{x^{2}}{1+x^{4}} d x\) (c) \(\int_{-\infty}^{\infty} \frac{d x}{1+x+x^{2}}\) (d) \(\int_{-\infty}^{\infty} \frac{x}{\left(x^{2}+2 x+2\right)^{2}} d x\) (e) \(\int_{-\infty}^{\infty} \frac{\cos a x}{1+x+x^{2}} d x\) (f) \(\int_{0}^{\infty} \frac{x^{2}+1}{x^{4}+1} d x\) (g) $\int_{0}^{\infty} \frac{x^{2}}{\left(x^{2}+4\right)^{2}\left(x^{2}+9\right)} d x$ (h) $\int_{0}^{\infty} \frac{x^{2}}{\left(x^{2}+1\right)\left(x^{2}+4\right)} d x$.

Chapter 9

Describe the singularity at \(z=\infty\) for the following functions. (a) \(\frac{2 z^{2}+1}{3 z^{2}-10}\) (b) \(\frac{z^{2}}{z+1}\) (c) \(\frac{z^{2}+10}{e^{z}}\) (d) \(\frac{e^{z}}{z^{2}+10}\) (e) \(\tan z-z\) (f) \(\frac{1}{z}+\sin z\).

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