Write sin(z) as the sum of a power series with centre Ti.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**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:**
- \( \sin(\pi i) = 0 \)
- \( \cos(\pi i) = \cosh(\pi) \)
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Transcribed Image Text:**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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