Proposition 2. (Comparison Test for improper integrals) Let f(t) and g(t) be functions on the domain (0,0). Suppose first that f(t) < g(t) for all t>a, and that g(t)dt converges. Then fa f(t)dt converges. Similarly, suppose that f(t) < g(t) for all t>a, and that f f(t)dt diverges. Then g(t)dt diverges. (You should think for a minute about why these should intuitively be true.) Problem 7: A) A function is of exponential type if there exists constants C, M and to with M> 0 not depending on t such that for all t > to, |ƒ(t)| < Met. Prove that if a function f(t) is piece-wise continuous on [0, ∞] and is of exponential type with constants M and C as above, then (L{f})(s) exists for all s > C. (Hint: first, get rid of a "finite" part of the integral, then use the above proposition.) B) Show that f(t) = et² is not of exponential type. (Hint: for any C>0, t² - Ct > 0 for all t> C. p(t) C) Show that for any C> 0 and any polynomial p(t), lim = 0. Note that this implies that p(x) is of exponential type. t-x Problem 8: Show that if f(t) is a function of exponential type (i.e. the Laplace Transform of f(t) actually exists), then (L{f'(t)})(s) = s(L{f})(s) — ƒ(0). (Hint: integration by parts.)
Proposition 2. (Comparison Test for improper integrals) Let f(t) and g(t) be functions on the domain (0,0). Suppose first that f(t) < g(t) for all t>a, and that g(t)dt converges. Then fa f(t)dt converges. Similarly, suppose that f(t) < g(t) for all t>a, and that f f(t)dt diverges. Then g(t)dt diverges. (You should think for a minute about why these should intuitively be true.) Problem 7: A) A function is of exponential type if there exists constants C, M and to with M> 0 not depending on t such that for all t > to, |ƒ(t)| < Met. Prove that if a function f(t) is piece-wise continuous on [0, ∞] and is of exponential type with constants M and C as above, then (L{f})(s) exists for all s > C. (Hint: first, get rid of a "finite" part of the integral, then use the above proposition.) B) Show that f(t) = et² is not of exponential type. (Hint: for any C>0, t² - Ct > 0 for all t> C. p(t) C) Show that for any C> 0 and any polynomial p(t), lim = 0. Note that this implies that p(x) is of exponential type. t-x Problem 8: Show that if f(t) is a function of exponential type (i.e. the Laplace Transform of f(t) actually exists), then (L{f'(t)})(s) = s(L{f})(s) — ƒ(0). (Hint: integration by parts.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please explain each step of getting this answer clearly, Don't just show the answer.
![Proposition 2. (Comparison Test for improper integrals) Let f(t) and g(t) be functions on the
domain (0, 0). Suppose first that f(t) < g(t) for all t > a, and that º g(t)dt converges. Then
Sa f(t)dt converges. Similarly, suppose that f(t) < g(t) for all t > a, and that f(t)dt diverges.
Then g(t)dt diverges.
(You should think for a minute about why these should intuitively be true.)
Problem 7: A) A function is of exponential type if there exists constants C, M and to with M > 0
not depending on t such that for all t > to, |f(t)| < MeCt. Prove that if a function f(t) is piece-wise
continuous on [0, ∞] and is of exponential type with constants M and C as above, then (L{f})(s)
exists for all s > C. (Hint: first, get rid of a "finite" part of the integral, then use the above
proposition.)
B) Show that f(t) = e² is not of exponential type. (Hint: for any C > 0, t² – Ct > 0 for all
t >C.
C) Show that for any C > 0 and any polynomial p(t), lim 2 = 0. Note that this implies
that p(x) is of exponential type.
Problem 8: Show that if f(t) is a function of exponential type (i.e. the Laplace Transform of f(t)
actually exists), then (L{f'(t)})(s) = s(L{f})(s) – f(0). (Hint: integration by parts.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa71931eb-10f4-4ca2-8b48-8ed6038caf23%2Ff86b10d8-3d2f-4ec8-b0d8-f6c69e6ee665%2Ftr4xzjq_processed.png&w=3840&q=75)
Transcribed Image Text:Proposition 2. (Comparison Test for improper integrals) Let f(t) and g(t) be functions on the
domain (0, 0). Suppose first that f(t) < g(t) for all t > a, and that º g(t)dt converges. Then
Sa f(t)dt converges. Similarly, suppose that f(t) < g(t) for all t > a, and that f(t)dt diverges.
Then g(t)dt diverges.
(You should think for a minute about why these should intuitively be true.)
Problem 7: A) A function is of exponential type if there exists constants C, M and to with M > 0
not depending on t such that for all t > to, |f(t)| < MeCt. Prove that if a function f(t) is piece-wise
continuous on [0, ∞] and is of exponential type with constants M and C as above, then (L{f})(s)
exists for all s > C. (Hint: first, get rid of a "finite" part of the integral, then use the above
proposition.)
B) Show that f(t) = e² is not of exponential type. (Hint: for any C > 0, t² – Ct > 0 for all
t >C.
C) Show that for any C > 0 and any polynomial p(t), lim 2 = 0. Note that this implies
that p(x) is of exponential type.
Problem 8: Show that if f(t) is a function of exponential type (i.e. the Laplace Transform of f(t)
actually exists), then (L{f'(t)})(s) = s(L{f})(s) – f(0). (Hint: integration by parts.)
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