**Problem Statement:**Write sin(z) as the sum of a power series with center πi.**Solution:**To express the function $ \sin(z) $ as a power series centered at $ \pi i $, we utilize the Taylor series expansion:1. **General Form:** The Taylor series for a function $ f(z) $ centered at $ z_0 $ is given by: \[ f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z-z_0)^n \]2. **Applying to $ \sin(z) $:** To find the power series for $ \sin(z) $ centered at $ \pi i $, we need the derivatives of $ \sin(z) $ evaluated at $ \pi i $. - **First Derivative:** $ \sin'(z) = \cos(z) $ - **Second Derivative:** $ \sin''(z) = -\sin(z) $ - **Third Derivative:** $ \sin'''(z) = -\cos(z) $ - **Fourth Derivative:** $ \sin^{(4)}(z) = \sin(z) $ (repeats every four terms)3. **Calculating Terms:** The specific terms require evaluation of these derivatives at $ z = \pi i $. - **0th Term:** $ \sin(\pi i) $ - **1st Term:** $ \frac{\cos(\pi i)}{1!} (z - \pi i) $ - **2nd Term:** $ \frac{-\sin(\pi i)}{2!} (z - \pi i)^2 $ - **3rd Term:** $ \frac{-\cos(\pi i)}{3!} (z - \pi i)^3 $ - **4th Term:** $ \frac{\sin(\pi i)}{4!} (z - \pi i)^4 $ - Continue this pattern for subsequent terms.4. **Trigonometric Evaluation:** - $ \sin(\pi i) = 0 $ - $ \cos(\pi i) = \cosh(\pi) $5.

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Answered: Write sin(z) as the sum of a power…

Math

Advanced Math

Write sin(z) as the sum of a power series with centre Ti.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Problem Statement:**

Write sin(z) as the sum of a power series with center πi.

**Solution:**

To express the function $ \sin(z) $ as a power series centered at $ \pi i $, we utilize the Taylor series expansion:

1. **General Form:**

The Taylor series for a function $ f(z) $ centered at $ z_0 $ is given by:

\[
f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z-z_0)^n
\]

2. **Applying to $ \sin(z) $:**

To find the power series for $ \sin(z) $ centered at $ \pi i $, we need the derivatives of $ \sin(z) $ evaluated at $ \pi i $.

- **First Derivative:** $ \sin'(z) = \cos(z) $
- **Second Derivative:** $ \sin''(z) = -\sin(z) $
- **Third Derivative:** $ \sin'''(z) = -\cos(z) $
- **Fourth Derivative:** $ \sin^{(4)}(z) = \sin(z) $ (repeats every four terms)

3. **Calculating Terms:**

The specific terms require evaluation of these derivatives at $ z = \pi i $.

- **0th Term:** $ \sin(\pi i) $
- **1st Term:** $ \frac{\cos(\pi i)}{1!} (z - \pi i) $
- **2nd Term:** $ \frac{-\sin(\pi i)}{2!} (z - \pi i)^2 $
- **3rd Term:** $ \frac{-\cos(\pi i)}{3!} (z - \pi i)^3 $
- **4th Term:** $ \frac{\sin(\pi i)}{4!} (z - \pi i)^4 $
- Continue this pattern for subsequent terms.

4. **Trigonometric Evaluation:**

- $ \sin(\pi i) = 0 $
- $ \cos(\pi i) = \cosh(\pi) $

5.

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