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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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![### Complex Integration Example
#### Problem:
Evaluate the contour integral:
\[ \int_C \frac{z}{z+1} \, dz \]
for a given contour \( C(t) \) where \( C(t) = 2e^{it} \) and \( t \in [0, \pi] \).
#### Solution:
1. **Given Contour:**
\[ C(t) = 2e^{it}, \quad t \in [0, \pi] \]
2. **Graphical Representation:**
The given contour describes a semicircle in the complex plane with a radius of 2 and centered at the origin, starting from \(2\) on the real axis and ending at \(-2\) on the real axis.
\[
\begin{array}{c}
\text{Diagram:}
\begin{array}{|c|c|}
\hline
\text{0} & 1 \\
\hline
\end{array}
\end{array}
\quad
\begin{array}{|c|}
\hline
\frac{2e^{it}}{2e^{it}+1} \\
\hline
\end{array}
\]
3. **Parametric Representation:**
\[ C(t) = \gamma(t) = 2e^{it} = 2(\cos t + i \sin t) = 2 \cos t + 2i \sin t \]
4. **Derivative of the Parametric Equation:**
\[ \gamma'(t) = 2i e^{it} \]
5. **Substitute \( f(z) = \frac{z}{z+1} \) into the integral:**
\[
f(\gamma(t)) = \frac{2e^{it}}{2e^{it} + 1}
\]
6. **Rewriting the integral:**
\[
\int_0^\pi \frac{2e^{it}}{2e^{it}+1} (2ie^{it}) \, dt
\]
7. **Simplify the integral:**
\[
= \int_0^\pi \frac{4ie^{2it}}{2e^{it} + 1}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f812bf4-4e37-4075-b786-04672d25a0a0%2Ffc721b03-6634-45b2-b464-d5e886a073ce%2Fxjqs8ze_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Complex Integration Example
#### Problem:
Evaluate the contour integral:
\[ \int_C \frac{z}{z+1} \, dz \]
for a given contour \( C(t) \) where \( C(t) = 2e^{it} \) and \( t \in [0, \pi] \).
#### Solution:
1. **Given Contour:**
\[ C(t) = 2e^{it}, \quad t \in [0, \pi] \]
2. **Graphical Representation:**
The given contour describes a semicircle in the complex plane with a radius of 2 and centered at the origin, starting from \(2\) on the real axis and ending at \(-2\) on the real axis.
\[
\begin{array}{c}
\text{Diagram:}
\begin{array}{|c|c|}
\hline
\text{0} & 1 \\
\hline
\end{array}
\end{array}
\quad
\begin{array}{|c|}
\hline
\frac{2e^{it}}{2e^{it}+1} \\
\hline
\end{array}
\]
3. **Parametric Representation:**
\[ C(t) = \gamma(t) = 2e^{it} = 2(\cos t + i \sin t) = 2 \cos t + 2i \sin t \]
4. **Derivative of the Parametric Equation:**
\[ \gamma'(t) = 2i e^{it} \]
5. **Substitute \( f(z) = \frac{z}{z+1} \) into the integral:**
\[
f(\gamma(t)) = \frac{2e^{it}}{2e^{it} + 1}
\]
6. **Rewriting the integral:**
\[
\int_0^\pi \frac{2e^{it}}{2e^{it}+1} (2ie^{it}) \, dt
\]
7. **Simplify the integral:**
\[
= \int_0^\pi \frac{4ie^{2it}}{2e^{it} + 1}
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