#Fourier Series Help needed

24 messages · Page 1 of 1 (latest)

torpid pine
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This is a class example I'm going over again, but I keep getting stuck finding the complex coefficient.

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prime sailBOT
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daring spade
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Deam

torpid pine
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This is what ive tried

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And this is how he did it in class, but im not sure how he got to his final answer

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proven rivet
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Which are all omitted

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I will try to detail the steps below

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\begin{align*}
\int_{0}^{1} (1 - t^2) \exp(-i\pi n t)dt &= \left[(1-t^2) \frac{\exp(-i\pi n t)}{-i \pi n} \right]_{0}^{1} - \int_{0}^{1} 2t \frac{\exp(-i\pi n t)}{-i \pi n} dt \\
&= \left[(1-t^2) \frac{\exp(-i\pi n t)}{-i \pi n} \right]_{0}^{1} - \left[2t \frac{\exp(-i\pi n t)}{(-i \pi n)^2} \right]_{0}^{1}\\ &+ \int_{0}^{1} 2 \frac{\exp(-i\pi n t)}{(-i \pi n)^2} \\
&= \left[(1-t^2) \frac{\exp(-i\pi n t)}{-i \pi n} \right]_{0}^{1} - \left[2t \frac{\exp(-i\pi n t)}{(-i \pi n)^2} \right]_{0}^{1} \\
 + \left[ 2 \frac{\exp(-i\pi n t)}{(-i \pi n)^3}\right]_{0}^{1}
\end{align*}
past kernelBOT
proven rivet
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Putting it all together under one pair of brackets:
$$\int_{0}^{1} (1 - t^2) \exp(-i\pi n t)dt = \left[\exp(-i\pi n t) \left(\frac{1-t^2}{-i\pi n} - \frac{2t}{(-i \pi n)^2} + \frac{2}{(-i \pi n)^3} \right) \right]_{0}^{1 }$$
past kernelBOT
proven rivet
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The rest just follows from simplifying all that

proven rivet
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1/i can be turned into -i

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(see that 1/i = i/i² = i/(-1) = -i)

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and normally you should find what the key says

torpid pine
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thank you, ill give it a shot, appreciate the help

proven rivet
torpid pine
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+close

fallen needleBOT
# torpid pine +close
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