(10t-24€+ /6) (1+32) dt 2. (10t-244 +16)12 +3i(10t3-24€+16) "EHE
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Integration Problem: Complex Function on a Contour
The following notes provide a detailed step-by-step solution to an integration problem involving a complex function over a specific contour \(C\).
#### Given Function and Interval
\[ \gamma(t) = t + (3t - 4)i \quad \text{for} \quad 1 \le t \le 2 \]
#### Problem Statement
(b) Integrate \(|z|\) over \(C\).
#### Integral Representation
\[ \int_C f = \int_a^b f(\gamma(t)) \gamma'(t) \, dt \]
Where
\[ f(z) = |z| \]
#### Detailed Steps:
1. **Parameterize the Function:**
\[ \gamma(t) = t + (3t - 4)i \]
for \( 1 \leq t \leq 2 \).
2. **Find the Derivative:**
\[ \gamma'(t) = 1 + 3i \]
3. **Determine \(f(\gamma(t))\):**
Since \(f(z) = |z|\),
\[ f(\gamma(t)) = |t + (3t - 4)i| \]
4. **Simplify the Magnitude:**
Let \( z = a + bi \)
\[
a = t
\]
\[
b = 3t - 4
\]
The magnitude \( |z| \) is:
\[
|z| = \sqrt{a^2 + b^2}
\]
Substitute \(a\) and \(b\):
\[
a^2 = t^2
\]
\[
b^2 = (3t - 4)^2 = 9t^2 - 24t + 16
\]
\[
|z| = \sqrt{t^2 + 9t^2 - 24t + 16} = \sqrt{10t^2 - 24t + 16}
\]
5. **Integral Formulation:**
Substitute back into the integral
\[
\int_1^2 \left( \sqrt{10t^2 - 24t + 16} \right)(1 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f812bf4-4e37-4075-b786-04672d25a0a0%2Ff97d4414-1470-4813-a251-0423c4c1cb6a%2Ffnuo45_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Integration Problem: Complex Function on a Contour
The following notes provide a detailed step-by-step solution to an integration problem involving a complex function over a specific contour \(C\).
#### Given Function and Interval
\[ \gamma(t) = t + (3t - 4)i \quad \text{for} \quad 1 \le t \le 2 \]
#### Problem Statement
(b) Integrate \(|z|\) over \(C\).
#### Integral Representation
\[ \int_C f = \int_a^b f(\gamma(t)) \gamma'(t) \, dt \]
Where
\[ f(z) = |z| \]
#### Detailed Steps:
1. **Parameterize the Function:**
\[ \gamma(t) = t + (3t - 4)i \]
for \( 1 \leq t \leq 2 \).
2. **Find the Derivative:**
\[ \gamma'(t) = 1 + 3i \]
3. **Determine \(f(\gamma(t))\):**
Since \(f(z) = |z|\),
\[ f(\gamma(t)) = |t + (3t - 4)i| \]
4. **Simplify the Magnitude:**
Let \( z = a + bi \)
\[
a = t
\]
\[
b = 3t - 4
\]
The magnitude \( |z| \) is:
\[
|z| = \sqrt{a^2 + b^2}
\]
Substitute \(a\) and \(b\):
\[
a^2 = t^2
\]
\[
b^2 = (3t - 4)^2 = 9t^2 - 24t + 16
\]
\[
|z| = \sqrt{t^2 + 9t^2 - 24t + 16} = \sqrt{10t^2 - 24t + 16}
\]
5. **Integral Formulation:**
Substitute back into the integral
\[
\int_1^2 \left( \sqrt{10t^2 - 24t + 16} \right)(1 +
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