Re: integrals involving many sines and cosines

rembremading <rembremading@xxxxxxx> writes:

Hi everybody!

Is there a general way to evaluate integrals of the form

\int_0^\infty ( sin(x*a_1)^e_1 ... sin(x*a_n)^e_n cos(x*b_1)^f_1 ...
cos(x*b_m)^f_m ) / x^\nu dx

where all the a's, b's are positive real numbers and the e's and f's are
positive integers?
For which \nu's does the integral converge and when not?
I could find integrals involving up to three sin's in the formulary.

Hints for convergence...
The integrand behaves like a constant times a power of x near x = 0.
What about as x -> infinity?

If you want the actual value, you can express those products of trig
functions as sums of trig functions, and use the integration-by-parts
reduction formula

int f(x)/x^n dx = -f(x)/((n-1) x^(n-1)) + 1/(n-1) int f'(x)/x^(n-1) dx

to eventually get it down to an integral where the denominator is x.
That can be evaluated using the Si and Ci functions.

Of course the simplest way is to use a Computer Algebra System such
as Maple or Mathematica.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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