An integral on the real line that spits out complex answer?

[Kurome]

I've recently encountered a baffling integral given by:

$$\int_0^\infty \exp \left(-x \sinh(t)-\frac{1}{2} t \right) dt$$​

Where x is restricted to be real.

I've tried doing the elementary methods like integration by parts or looking for differentials, but with no luck. I was quite baffled when I tried to use a computer algebra system to compute it. It's spitting out complex numbers and special functions (imaginary error function and complementary error function).

I know this definitely isn't your usual integral, it's actually one of the terms of Schafli's integral representation of the Bessel function with parameter 1/2.

Can somebody help me evaluate this? I'm thinking of studying how Schafli's integral representation of that Bessel integral was derived (I know it involves countour integrals), but I'm not quite sure if it will give me a clue on how to solve the integral above. I'm not even sure if this can be solved analytically.

 

[jvicens]

Thank you for sharing your experience with this integral. It is indeed a challenging one, and I can understand your frustration with trying to evaluate it using elementary methods.

As you mentioned, this integral is part of Schafli's integral representation of the Bessel function with parameter 1/2. This representation involves contour integration, and it is not surprising that a computer algebra system would give complex numbers and special functions as the result.

To evaluate this integral, you will need to use more advanced methods such as contour integration, residue theorem, and the theory of special functions. These methods are beyond the scope of elementary calculus, but they are commonly used in advanced mathematics and physics.

I would suggest studying Schafli's integral representation and the theory of special functions to gain a better understanding of this integral. This will also give you a better idea of how to approach similar integrals in the future.

I hope this helps, and I wish you all the best in your studies. Don't get discouraged, as challenging integrals like this one often lead to new insights and discoveries in mathematics.