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Results tagged with integration
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user 8955
Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...
3
votes
Accepted
Integral with confluent hypergeometric function
Hope this helps some,
Tom
Did this integral come from using Mellin transforms + Parseval to perform an integration? …
- 1,077
1
vote
Numerical multivariate definite integration
Look up "gaussian integral" on Wikipedia. I think you can handle the absolute value portion by splitting the range, and you will get the integral in terms of $ \det A$. Take a look.
Tom
- 1,077
3
votes
Accepted
Contour Integral with Gamma functions and 2F1
Then you need to estimate the behavior of your integrand for large $\Im(s) $, to determine which direction you could move the integration contour and pick up poles to develop a series for your integral …
- 1,077
3
votes
Accepted
Definite integral with modified Bessel functions, trigonometric function and a power
One can use Mellin transforms to tackle this integral; in this case one obtains series involving hypergeometric functions. Here is a very brief summary of the process. I plan to complete the answer wi …
- 1,077
3
votes
An integral that somehow equals pi^2/6 and involves dilogarithms?
This problem can also be approached by rewriting the sum. I'll show a lot of details, probably too many, here.
Use the binomial series to write
$ \frac{1}{(1+x^k)^2} = \sum_{m=0}^{\infty} \binom{m+1 …
- 1,077
4
votes
Best Numerical Method for Evaluating a Hilbert transform
There is a very recent paper in Mathematics of Computation,
"Computing the Hilbert transform and its inverse"
Sheehan Olver
Math. Comp. 80 (2011), 1745-1767.
He presents a new algorithm and refer …
- 1,077
5
votes
How to do integrals involving two Bessel functions and another function?
Take a look at "A Treatise on the Theory of Bessel Functions" by Watson. There is a long chapter on integrating Bessel functions over the infinite range $0-\infty$.
In addition, I think a Mellin tra …
- 1,077
3
votes
Is there a closed form expression/series expansion for $\int_{\epsilon-i\infty}^{\epsilon+i\...
Since this integral has no
singularity on the real axis (please note, I have changed variables using
$z = i u$, so the integration is along the real axis),
we may take $\epsilon=0$ and write
$$
I_2 …
- 1,077