I'm trying but no luck. Can't find a pattern yet. The exercise is to find the nth derivative of $e^{-x}\sin(x)$ probably by induction.
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Is the sine really in the exponent? (I would have expected $e^{-x}\sin x$ to which one could at least apply Leibniz' rule for $n$th derivative of a product.) – coffeemath Sep 25 '14 at 23:31
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The edit is wrong. – Clarinetist Sep 25 '14 at 23:32
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Which edit is wrong? Now it looks like what I thought it should be (last comment), but it would be good of Gonzalo were to confirm the final version of the function. A minute or two ago it had the sine in the exponent, at my site. – coffeemath Sep 25 '14 at 23:37
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1Look at the original post. http://math.stackexchange.com/posts/946280/revisions – Clarinetist Sep 25 '14 at 23:39
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I did it for $e^x \sin x$ here, it might be helpful. Try to mimic it. – Ivo Terek Sep 26 '14 at 00:17
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Thanks, is the i just a normal variable or i as in sqrt(-1) ? Also the sine isn't the exponent, sorry this was my first question here and i'm note really familiar with it's format. – Gonzalo Fernández Sep 26 '14 at 00:20
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Hehe http://en.m.wikipedia.org/wiki/General_Leibniz_rule – ClassicStyle Sep 26 '14 at 01:50
2 Answers
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We have $\sin z = \Im(e^{iz})$, hence: $$\frac{d^n}{dz^n}\left(e^{-z}\sin z\right)=\Im\left(\frac{d^n}{dz^n}e^{(i-1)z}\right)=\Im\left((i-1)^n e^{(i-1)z}\right)=2^{n/2}e^{-z}\,\Im\left(e^{i(z+3n\pi/4)}\right)$$ giving: $$\frac{d^n}{dz^n}\left(e^{-z}\sin z\right)=2^{n/2}e^{-z}\sin(z+3\pi n/4).$$
Jack D'Aurizio
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Thanks for the answer however I think I lack the knowledge to understand it. Why would sinz=integral of e^(iz) ? – Gonzalo Fernández Sep 26 '14 at 00:54
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1@GonzaloFernández: $\Im$ stands for the imaginary part, not the integral. – Jack D'Aurizio Sep 26 '14 at 01:00
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Oh I see. Then i definitely lack the knowledge to comprehend this. Thank you anyways ;) – Gonzalo Fernández Sep 26 '14 at 01:02
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Perhaps this is an easier way. Put $g(x)= e^{(-1+i)x}$ and note that your $f(x)$ is the imaginary part of $g(x)$ for real $x$. Now $g^{(n)}(x) =(-1+i)^n e^{(1+i)x}$.Now i am sure you can proceed further!
Arpit Kansal
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