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goldfish9776
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Homework Statement
for the alternative form of second shift property (4.8) , why he integral of (e^-sp) g(p+a) dp isn't equal to integral of (e^-sp) g(t) dp ? why it will become L{ g(t+a) } ?
ya , why not the final ans not L {g(p+a)} ?RUber said:Would you mind showing more work explaining your reasoning?
I imagine the answer is either in the endpoints of your integrals or the substitution.
Straight off, from your question, for ##t = p+a##
##\int e^{-sp}g(p+a) dp = \int e^{-sp}g(t) dp##
but you can't do much with that...complete the substitution to change the whole integral into one with only dependence on t and no p's.
i gt stucked here , how to continue?RUber said:It looks like you are treating g(p+a) as a multiplication. g(x) is a function.
Also, when you do substitutions within integrals, you have to change the limits as well.
For example:
Substituting ##t = p+a\quad p = t-a\quad p=0 \implies t = a##
##\int_0^\infty e^{-sp}g(p+a) dp = \int_{a}^\infty e^{-s(t-a)}g(t) dt##
Does that make sense?
Two things I noticed in your post #7...goldfish9776 said:i gt stucked here , how to continue?
The Second Shift Theorem is a mathematical theorem used to evaluate definite integrals with trigonometric functions. It allows for the shifting of the function's argument by a constant value, making the integration process simpler.
The Second Shift Theorem is used by substituting the argument of the function with a new variable. This variable is then shifted by a constant value, which allows for the use of simpler trigonometric identities to solve the integral.
The formula for the Second Shift Theorem is ∫f(x)dx = ∫f(x+a)dx = ∫f(x)dx + a, where 'a' is the constant value used to shift the function's argument.
The Second Shift Theorem is useful because it simplifies the integration process for trigonometric functions. It allows for the use of simpler identities and reduces the number of steps required to solve the integral.
The Second Shift Theorem has various applications in fields such as physics, engineering, and mathematics. It is commonly used in the evaluation of Fourier series, solving differential equations, and calculating areas under curves.