Show that F(s) est ds if F(s) for ->O as $18 t>o
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
![**Mathematical Analysis**:
**Objective**: Demonstrate the behavior of an integral involving an exponential function.
**Problem Statement**:
Show that
\[
\int_{\Gamma_1} F(s) e^{st} \, ds \to 0
\]
for \( t > 0 \), if \( F(s) \to 0 \) as \( s \to \infty \).
**Explanation**:
- The integral is taken over a path \(\Gamma_1\).
- \( F(s) \) is a function of \( s \), which approaches zero as \( s \) approaches infinity.
- \( e^{st} \) represents an exponential function with respect to \( s \) and the constant \( t \).
- The goal is to show that this integral approaches zero under the given conditions.
**Graphical Representation**:
No graphs or diagrams are present or required in this presentation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71f6a4b8-527f-4030-b85a-828002701c6f%2F2972830e-1f73-4696-8d65-31750e3e18af%2Faa48oaq_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematical Analysis**:
**Objective**: Demonstrate the behavior of an integral involving an exponential function.
**Problem Statement**:
Show that
\[
\int_{\Gamma_1} F(s) e^{st} \, ds \to 0
\]
for \( t > 0 \), if \( F(s) \to 0 \) as \( s \to \infty \).
**Explanation**:
- The integral is taken over a path \(\Gamma_1\).
- \( F(s) \) is a function of \( s \), which approaches zero as \( s \) approaches infinity.
- \( e^{st} \) represents an exponential function with respect to \( s \) and the constant \( t \).
- The goal is to show that this integral approaches zero under the given conditions.
**Graphical Representation**:
No graphs or diagrams are present or required in this presentation.
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