Numerical integration with large initial limit

Questions
,
1

I’m facing an issue with the integral attached in the MA file.

Can Mathematica establish this integral with large integration initial limit (a =- 10^9) ? In the current MA file, the integration only works because I set the initial integration limit to (a=1).

I’m using MA 12:00, so can integration Method used or so?

Also, how to take the limit of the integration at q \to 0 ?

I mean one can take the limit of Integrate like:

Limit[Integrate[Exp[a x], {x, 1, q}], q -> 0]

but how to do so with NIntegrate ?

Thanks,
Safinaz Salem

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2

The integral i1 can be done symbolically:

Map[Integrate[#, {b, 1, q}] &, eq[k, q, b, f] //
   FullSimplify // Expand]
3

Hello, thanks a lot for your answer. You mean to try something like:

i1[k_, f_,q_]=Map[Integrate[#, {b, a, q}] &, eq[k, q, b, f] // FullSimplify // Expand]

i2[k_,q_]=Map[Integrate[#, {f, a, q}] &, i1[k,f,q] // FullSimplify // Expand]

But it takes so much time to be evaluated. Is there a way to speed up the integration ?

4

I only tried the i1. The i2 is probably too difficult for a symbolic solution.

5

I try to evaluate i2[k,q] in the limit where q\to0, so I guess to take The limit, i2 should be done by Integrate too, because how to take the limit in NIntegrate?

In the attached MA file, I evaluate i2[k, 0.01] to consider q is so small, but I don’t think this is accurate and one should take the Limit.

6

Your i2 can be written as a double integral on a variable square. Make a change of variable so that the integral become on a fixed square. Then find the limit of the function inside the interval, if possible. That may be a start.

7

Thanks a lot I will give it a try.