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Math Help - complex derivative

  1. #1
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    complex derivative

    The following problem appears in my complex analysis text and I was wondering how you solve it in this context.
    Find the 10th derivative of [e^(0.5x)]*sin((sqrt(3)/2)x) (obviously without taking the derivative ten times).

    Thanks for any input.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by PvtBillPilgrim View Post
    The following problem appears in my complex analysis text and I was wondering how you solve it in this context.
    Find the 10th derivative of [e^(0.5x)]*sin((sqrt(3)/2)x) (obviously without taking the derivative ten times).

    Thanks for any input.
    we still have to take quite a few derivatives from what i can see, but they should be easier as you can see a pattern.

    note that the derivative you seek is the real part of the 10th derivative of -i \text{exp}\left( \frac x2 + i\frac {\sqrt{3}}2x\right) which is the real part of -i \left( \frac 12 + i \frac {\sqrt{3}}2\right)^{10} \text{exp} \left( \frac x2 + i\frac {\sqrt{3}}2x\right) = -i   \text{exp} \left( i \frac {10 \pi}3\right) \cdot \text{exp} \left( \frac x2 + i\frac {\sqrt{3}}2x\right) =  -i  \text{exp} \left[ \frac x2 + i \left( \frac {10 \pi}3 + \frac {\sqrt{3}}2 \right) x\right]
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