L(sin Vf) = V. 2s
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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By taking the laplace transform of sin(√(t)) = t^(1/2) - ((t^(3/2))/3! +((t^(5/2))/5! - ... show what is depicted in the picture below containing laplace of sin(√(t))
![The image presents a mathematical expression involving the Laplace transform. The expression is as follows:
\[ \mathcal{L}(\sin \sqrt{t}) = \frac{\sqrt{\pi}}{2s^{\frac{3}{2}}} e^{-\frac{1}{4s}}. \]
This equation illustrates the Laplace transform of the function \(\sin \sqrt{t}\). Here, \(\mathcal{L}\) denotes the Laplace transform operator. The result of the transform involves several components:
- \(\frac{\sqrt{\pi}}{2s^{\frac{3}{2}}}\) is a multiplicative factor, where \(s\) is the complex frequency parameter used in Laplace transforms.
- \(e^{-\frac{1}{4s}}\) is an exponential function showing the decay factor related to \(s\).
This expression provides the transformed version of the time-domain function \(\sin \sqrt{t}\) in the s-domain, which is useful in many applications of engineering and physics for solving differential equations and analyzing systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40884399-19a4-4ad9-a712-30c31bac2ede%2F801f81a8-ba1b-48e7-9b1b-bf152bdc2f63%2Fa4nfhn6_processed.png&w=3840&q=75)
Transcribed Image Text:The image presents a mathematical expression involving the Laplace transform. The expression is as follows:
\[ \mathcal{L}(\sin \sqrt{t}) = \frac{\sqrt{\pi}}{2s^{\frac{3}{2}}} e^{-\frac{1}{4s}}. \]
This equation illustrates the Laplace transform of the function \(\sin \sqrt{t}\). Here, \(\mathcal{L}\) denotes the Laplace transform operator. The result of the transform involves several components:
- \(\frac{\sqrt{\pi}}{2s^{\frac{3}{2}}}\) is a multiplicative factor, where \(s\) is the complex frequency parameter used in Laplace transforms.
- \(e^{-\frac{1}{4s}}\) is an exponential function showing the decay factor related to \(s\).
This expression provides the transformed version of the time-domain function \(\sin \sqrt{t}\) in the s-domain, which is useful in many applications of engineering and physics for solving differential equations and analyzing systems.
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