Integrate problem with undefined constant

[sukharef]

Hello!

I'm trying to calculate this integral:

Integrate [Exp[-I a *y] Exp[I*Sin[y]] Exp[-y*Sign[y]], {y, -Infinity, Infinity}] , where "a" - constant that i'll define later.

So the result of the calculation acсording to Mathematica is the same expression : Exp[-I a *y] Exp[I*Sin[y]] Exp[-y*Sign[y]].

What can i do to calculate my integral without defining my constant "a" and using NIntagrate?

Thank you in advance.

 

[Simon_Tyler]

I doubt that Mma returns the integrand - it probably returns an unevaluated integral, no matter what choice you make for the constant a. The integral probably does not have a nice closed form in terms of any common or existing special functions.

Mma doesn't even evaluate the "simpler" integral Exp[I*Sin[y] - y^2].

If you just need to use this result just define the numerical integral (with memoization)

int[a_?NumericQ] := int[a] = NIntegrate[Exp[-I a*y] Exp[I*Sin[y]] Exp[-y*Sign[y]], {y, -Infinity, Infinity}]//Chop

It's a little slow... but you can get a reasonable plot of the integral

Plot[int[x], {x, -10, 10}, PlotPoints -> 15, MaxRecursion -> 3, PlotRange -> All]

 

[sukharef]

Thank you so much for answer!

 

[Simon_Tyler]

Not a problem. I hope it was useful.
If you do find a closed form to that integral - don't forget to post it here!

 

[alphy]

I understand the frustration of trying to solve a problem with an undefined constant. In this case, it seems like your integral is dependent on the value of "a", which makes it difficult to find a solution without defining it. One approach you could try is to use a symbolic integration method, such as the residue theorem, which does not require a specific value for "a". Another option is to find a way to express the integral in terms of a known function or series, such as a Taylor series, which may allow you to evaluate it without the need for a constant. Alternatively, you could consider setting a range of values for "a" and using numerical methods, such as NIntegrate, to approximate the integral for each value. This would give you a better understanding of how the integral behaves as "a" varies. I hope these suggestions help in solving your problem. Good luck!