San José State University |
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applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
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Integral of Functions |
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Consider integrals of the form
What is sought is the condition on F(x) such that J(λ)→0 as λ→∞. The analysis will start with F(x)=x^{n} for n as a nonnegative integer.
First of all, by letting λx=z
When Integration-by-Parts (IbP) is applied to ∫z^{n}sin(z)dz the result is
When this integral is evaluated at the limits of λL and λH and the results substituted into the transformed expression for I_{n}(λ) the result is
Both terms enclosed in square brackets on the RHS of the above are finite so the asymptotic limits of the first two terms on the RHS are zero. Now define K_{n} as
Then from the previous result
Thus if K_{m} = 0 then K_{m+2} = 0.
It is easily shown that K_{0}=0 and K_{1}=0. Therefore K_{n}=0 for all nonnegative n.
Thus for any function defined by a polynomial series
the integral J(λ) = ∫_{L}^{H}F(x)sin(λx)dx has an asymptotic limit of zero as λ increases without limit.
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