 Suggested languages for you:

Europe

Problem 2

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

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.
See the step by step solution

## 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 $$R \rightarrow \infty$$.

We value your feedback to improve our textbook solutions.

## Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features ## Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

• Flashcards & Quizzes
• AI Study Assistant
• Smart Note-Taking
• Mock-Exams
• Study Planner 