- #1

VantagePoint72

- 821

- 34

For example, suppose you wanted to do the following integral:

[tex]\int_{-\infty}^{\infty} e^{-\left(t + i\alpha\right)^2} dt[/tex]

with [itex]\alpha[/itex] real. Can I just do u-substitution like this:

[tex]u = t + i\alpha \rightarrow du = dt \\

\int_{-\infty}^{\infty} e^{-\left(t + i\alpha\right)^2} dt =

\int_{u=-\infty}^{u=\infty} e^{-u^2} du [/tex]

and conclude the integral is [itex]\sqrt{\pi}[/itex] like usual? Punching it into Wolfram Alpha confirms that's the result, so it's more whether or not what I did was valid in general that I'm interested in, rather than this particular result. Can I still use the familiar tricks, or does the mere presence of complex numbers invalidate these old techniques?