nth derivative

**harmonicadudee** · November 27th 2009, 01:07 PM

Show that

$\frac{d^n}{dx^n} (e^{ax}sinbx) = r^ne^{ax}sin(bx+n\theta)$

where $a$ and $b$ are positive numbers, $r^2 = a^2 + b^2$ and $\theta =tan^{-1}(b/a)$

**qmech** · November 27th 2009, 01:13 PM

Try using
$ e ^ {ix } = cos (x) + i sin (x) $
to write
$ sin (x) = Im ( e ^ {ix } ) $
where Im means "imaginary part of".

**harmonicadudee** · November 27th 2009, 02:02 PM

I really don't know much about imaginary numbers, so I don't know where to go from there. Is there possibly any other way to solve this problem?

**qmech** · November 27th 2009, 04:36 PM

Complex numbers are very useful. They also have some interesting properties. One is, if:
$ z = r ( cos (\theta) + i sin (\theta) ) $
then
$ z^n = r^n ( cos (n\theta) + i sin (n\theta) ) $

Check your text for a review on complex numbers.

**harmonicadudee** · November 27th 2009, 05:50 PM

ok i guess i'll try to learn about them and then i'll get back to you

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