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(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

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.3: The Addition And Subtraction Formulas
Problem 72E
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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 325² ²5² ² How does your result relate to the previous parts of this question?
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 325² ²5² 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: 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
Transcribed Image Text:(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
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