Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm trying to calculate this integral but I'm a bit stuck. Has anyone got any tips/tricks to deal with the $e^{ir\cosθ}$ part?

$$\iiint r^{2}e^{ir\cos\theta}\sin\theta \,dr\,d\theta \,d\phi$$

Limits: $0\leq r \leq a$, $0\leq\theta\leq \pi$, $0\leq\phi\leq2\pi$.

I'm a first year chemistry student so keep the maths as simple as possible!

share|cite|improve this question
up vote 3 down vote accepted

Your integral can be rewritten (by Fubini): $$ \left(\int_{\phi=0}^{2\pi}d\phi\right)\left(\int_{r=0}^ar^2\left(\int_{\theta=0}^\pi e^{ir\cos\theta}\sin\theta d\theta\right)dr\right) $$ Of course, the first factor is $2\pi$. Now for every $r>0$, do the change of variable $u=ir\cos\theta$, $du=-ir\sin\theta d\theta$ in the middle integral to get $$ \int_{\theta=0}^\pi e^{ir\cos\theta}\sin\theta d\theta=\int_{ir}^{-ir}e^u\frac{-du}{ir}=\frac{1}{ir}\int_{-ir}^{ir}e^udu=\frac{1}{ir} e^u\rvert_{-ir}^{ir}= \frac{1}{ir}(e^{ir}-e^{-ir}). $$ Now $$ \int_{r=0}^ar^2\left(\int_{\theta=0}^\pi e^{ir\cos\theta}\sin\theta d\theta\right)dr=\frac{1}{i}\int_0^ar(e^{ir}-e^{-ir})dr=2\int_0^ar\sin r dr $$ by Euler's formula $e^{ir}-e^{-ir}=2i\sin r$.

It only remains to integrate by parts $$ \int_0^ar\sin dr=(-r\cos r)\rvert_0^a+\int_0^a\cos r dr=-a\cos a+\sin r\rvert_0^a=-a\cos a+\sin a. $$ Finally, your integral is worth $$ 2\pi\cdot2(\sin a-a\cos a)=4\pi(\sin a -a\cos a). $$

share|cite|improve this answer
And you wonder why I put my $d$'s next to my integrals? – Ron Gordon Mar 14 '13 at 19:31
@RonGordon He he he...I see it now. – 1015 Mar 14 '13 at 19:33

Hint: $$(e^{\cos{t}})'=-\sin{t} \cdot e^{\cos{t}}$$ Integrate with respect to $\theta$ first, you should get $ir(e^{ir}-e^{-ir})$.

Use the formula $$sinr=\frac{e^{ir}-e^{-ir}}{2i}$$

You should be able to come up with the rest.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.