(2) Laplace transform of hyperbolic functions. The hyperbolic functions are defined as follows: et +e-x ex-e-x sinh(x):= 2 These function appear in math (and real life) on various occasions, including in the formula for computing the interval] in special relativity. (a) Show by using Euler's formula that cos(it) = cosh(t) and sin(it) = i-sinh(t), namely the hyperbolic functions give meaning to evaluating the trigonometric functions with imaginary arguments. Also show that cos(t) = cosh(it) and sin(t) = −i - sinh(it). (b) Explain why if a > 0 is a constant, sinh(a - t) and cosh(at) actually should have Laplace transforms to begin with (a question you should be able to answer without trying to actually find the transforms). Then verify the following formulas for their respective Laplace transforms. L{cosh(a-t)} = S s2_a² cosh(x):= et te 2 Z{sinh(a-t)} = a s²_a²
(2) Laplace transform of hyperbolic functions. The hyperbolic functions are defined as follows: et +e-x ex-e-x sinh(x):= 2 These function appear in math (and real life) on various occasions, including in the formula for computing the interval] in special relativity. (a) Show by using Euler's formula that cos(it) = cosh(t) and sin(it) = i-sinh(t), namely the hyperbolic functions give meaning to evaluating the trigonometric functions with imaginary arguments. Also show that cos(t) = cosh(it) and sin(t) = −i - sinh(it). (b) Explain why if a > 0 is a constant, sinh(a - t) and cosh(at) actually should have Laplace transforms to begin with (a question you should be able to answer without trying to actually find the transforms). Then verify the following formulas for their respective Laplace transforms. L{cosh(a-t)} = S s2_a² cosh(x):= et te 2 Z{sinh(a-t)} = a s²_a²
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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Transcribed Image Text:**Note:** You can actually do this *without evaluating any integrals,* but possibly using some more elementary facts about Laplace transforms. Please prove your answer.
**(c)** Calculate (directly, by means of partial-fraction decomposition) the inverse Laplace transforms of \(\frac{s}{s^2-a^2}, \frac{a}{s^2-a^2}\). How does your result relate to the previous parts of this question?
![**(2) Laplace transform of hyperbolic functions.** The hyperbolic functions are defined as follows:
\[
\sinh(x) := \frac{e^x - e^{-x}}{2}, \quad \cosh(x) := \frac{e^x + e^{-x}}{2}.
\]
These functions appear in math (and real life) on various occasions, including in the formula for computing the *interval* in special relativity.
(a) Show by using Euler’s formula that \(\cos(it) = \cosh(t)\) and \(\sin(it) = i \cdot \sinh(t)\), namely the hyperbolic functions give meaning to evaluating the trigonometric functions with imaginary arguments. Also show that \(\cos(t) = \cosh(it)\) and \(\sin(t) = -i \cdot \sinh(it)\).
(b) Explain why if \(a > 0\) is a constant, \(\sinh(a \cdot t)\) and \(\cosh(a \cdot t)\) actually should have Laplace transforms to begin with (a question you should be able to answer without trying to actually find the transforms). Then verify the following formulas for their respective Laplace transforms.
\[
\mathcal{L}\{\cosh(a \cdot t)\} = \frac{s}{s^2 - a^2}, \quad \mathcal{L}\{\sinh(a \cdot t)\} = \frac{a}{s^2 - a^2}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ea01508-f9b6-4a19-b038-56e12c298daf%2Fb4e0174e-e447-4b1f-bd7f-53b16ef9fbb4%2Foanq3np_processed.png&w=3840&q=75)
Transcribed Image Text:**(2) Laplace transform of hyperbolic functions.** The hyperbolic functions are defined as follows:
\[
\sinh(x) := \frac{e^x - e^{-x}}{2}, \quad \cosh(x) := \frac{e^x + e^{-x}}{2}.
\]
These functions appear in math (and real life) on various occasions, including in the formula for computing the *interval* in special relativity.
(a) Show by using Euler’s formula that \(\cos(it) = \cosh(t)\) and \(\sin(it) = i \cdot \sinh(t)\), namely the hyperbolic functions give meaning to evaluating the trigonometric functions with imaginary arguments. Also show that \(\cos(t) = \cosh(it)\) and \(\sin(t) = -i \cdot \sinh(it)\).
(b) Explain why if \(a > 0\) is a constant, \(\sinh(a \cdot t)\) and \(\cosh(a \cdot t)\) actually should have Laplace transforms to begin with (a question you should be able to answer without trying to actually find the transforms). Then verify the following formulas for their respective Laplace transforms.
\[
\mathcal{L}\{\cosh(a \cdot t)\} = \frac{s}{s^2 - a^2}, \quad \mathcal{L}\{\sinh(a \cdot t)\} = \frac{a}{s^2 - a^2}.
\]
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