How to evaluate the integral of the exponential of tangent squared?

Question

I was stopped during evaluation of other problem by the integral of
$$int_0^frac pi 2 e^{ictan^2theta}dtheta$$ where $cneq 0$ and $cinRe$.
By variable substitution $x=tantheta$, it becomes equivalently
$$int_0^inftyfrac{e^{icx^2}}{1+x^2}dx.$$ Seems not direct to evaluate and turned to complex plane with two poles $z=pm i$ and the results vanish. But not sure if this is correct?

Claude Leibovici · Accepted Answer

It is a bit tedious but workable using partial fraction decomposition.
For each integral appear the cosine and sine integrals as well as the $text{erf}(.)$ function (this looks normal to me).
After simplifications, the result seems to be
$$int_0^inftyfrac{e^{icx^2}dx}{1+x^2},dx=frac{pi}{2}   e^{-i c} left(1+i, text{erfi}left((1+i)
   sqrt{frac{c}{2}}right)right)$$ which efffectively tends to $0$ when $c to infty$.
A few values for $c=10^k$
$$left(
begin{array}{cc}
 k & text{result} \
 0 & 0.6528042545, + ,0.3617854763 , i \
 1 & 0.2063689904, + ,0.1872057102 , i \
 2 & 0.0629742223, + ,0.0623477998 , i \
 3 & 0.0198265299, + ,0.0198067133 , i \
 4 & 0.0062668840, + ,0.0062662573 , i \
 5 & 0.0019816736, + ,0.0019816537 , i \
 6 & 0.0006266574, + ,0.0006266568 , i \
 7 & 0.0001981664, + ,0.0001981664 , i \
 8 & 0.0000626657, + ,0.0000626657 , i \
 9 & 0.0000198166, + ,0.0000198166 , i
end{array}
right)$$
These results have been checked using numerical integration.
Using Mathematica, the result is confirmed if $Im(c)>0$
Edit
For large values of $c$ the expansion is
$$frac{1+i}{2} sqrt{frac{pi }{2c}}left(1-frac{i}{2 c}-frac{3}{4 c^2}+frac{15 i}{8
   c^3}+frac{105}{16 c^4}-frac{945
   i}{32 c^5}+Oleft(frac{1}{c^6}right)right)$$ For $c=10$, this would give
$$frac{666609,+,604631, i}{2560000} sqrt{frac{pi }{5}}approx 0.2064054,+, 0.1872149,i$$

How to evaluate the integral of the exponential of tangent squared?

One Answer

Add your own answers!

Ask a Question