A complex definite integral used in the semi-analytical solution of the mass transport equation in an unbounded two- or three-dimensional domain (Q1191779)

The author calculates the integral \[ \int^ \infty_ 0 e^{-(t^ 2+{z^ 2\over t^ 2})}dt={\sqrt\pi\over 2}e^{-2z},\quad|\arg z|\leq{\pi\over 4}. \] For positive \(z\) this equality is reported in mathematical tables. For complex \(z\) this equality follows from the theorem of uniqueness for analytic functions. The author gives another proof.