Is the Substitution Rule Applicable to Complex Numbers?

[pwsnafu]

I was reading J. J. Koliha's book on analysis and came across this

It is important to bear in mind that the substitution operates from R to R; the temptation to substitute u(t) = sin t + 3i in \int_0^{\pi/2}(\sin t + 3i)^{-2} \cos t \, dt must be resisted. The right substitution is u(t) = sin t.

Do you agree?

 

[snipez90]

No. It's intuitively obvious to me that the answer should be 'No'. The first justification I could think of was complex line integrals and how a fundamental theorem of calculus analogue for them can be proved easily.

 

[HilbertSpace]

The substitution would certainly be effective in solving the integral.

If you make the substitution u=sint, then du=cost dx, so 1/(cost) du = dx

Therefore the integral simplifies to integrating (u+3i)^(-2) du.

 

[theorem4.5.9]

Pay close attention to the quoted text. A substitution like that is indeed is only valid when your substitution is a function from $\mathbb{R}$ to $\mathbb{R}$.

In the complex sense, you have to be sure your substitution avoids poles (singularities) in the complex plane. For the real $t$, $\mathrm{sin}(t)$ is never zero. That's probably not the case if $t$ were complex.