Integration Techniques - Trigonometric Function Substitutions - cis(x)

[GreenPrint]

Hi,

I was wondering, just out of curiosity, what family of integrands can be evaluated by making a cis(x) substitution, cis(x)=cos(x)+isin(x), I can't seem to find any examples or anything at all and was wondering if someone could provide an example or inform of how to make such substitutions.

 

[Gib Z]

There isn't really a protypical situation to use $$u = \cos x + i \sin x$$ like there are for other substitution, eg trying $t=\tan x$ if one sees $$\sqrt{1+t^2}$$ in the integrand. Thing will become for clear when you learn that this mysterious function cis(x) is actually just $$e^{ix}$$ and all the rules you've learned about exponentials before will apply again.

 

[GreenPrint]

Ya I understand and am familiar with the function. The only problem is that I can't think of a situation period in which I could make such a substitution, nor can I find any examples of when I could do so.

 

[Screwdriver]

Actually, I remember reading something relevant to this in J. Nearing's Mathematical Tools for Physics. Consider:

$$\int cos(ax)e^{bx}dx + i \int sin(ax)e^{bx}dx$$

Then by adding integrands:

$$\int e^{bx}[cos(ax) + i \, sin(ax)]dx$$

Here we make the substitution $cos(ax) + i \, sin(ax) = cis(ax) = e^{i(ax)}$

$$\int e^{bx} e^{iax} dx$$
$$\int e^{(b + ia)x} dx$$

Now the integral is extremely easy. After evaluation, one can consider the real and imaginary parts to obtain $\int cos(ax)e^{bx}dx$ and $\int sin(ax)e^{bx}dx$ without integrating by parts even once!

Try finding that in a Biology textbook.

 

[GreenPrint]

Hmm... Very interesting. Thanks. That's a great integration technique.

 

[Screwdriver]

Don't spread it around too much