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Variants of the Clausen Functions

To Prove: \[\sum _{ k=1 }^{ \infty }{ \frac { \cos { \left( kx \right) } }{ { k }^{ 2 } } } =\frac { { \pi }^{ 2 } }{ 6 } -\frac { \pi x }{ 2 } +\frac { x^{ 2 } }{ 4 } \]

\(0\leq x\leq 2\pi\)

Proof:

Write \(\cos(kx)\) as: \(\displaystyle \cos { \left( kx \right) } =\frac { { e }^{ ikx }+{ e }^{ -ikx } }{ 2 } \)

Then we get \[S=\sum _{ k=1 }^{ \infty }{ \frac { \frac { { e }^{ ikx }+{ e }^{ -ikx } }{ 2 } }{ { k }^{ 2 } } } \]

Then on using the definition of polylogarithm I'll write it as: \[S=\frac { { Li }_{ 2 }\left( { e }^{ ikx } \right) +{ Li }_{ 2 }\left( { e }^{ -ikx } \right) }{ 2 } \]

Now, I'll use the following relation of polylogarithm and bernoulli numbers: \[{ Li }_{ m }(z)\quad =\quad { (-1) }^{ m-1 }{ Li }_{ m }\left( \frac { 1 }{ z } \right) -\frac { { (2\pi i) }^{ m } }{ m! } { B }_{ m }\left( \frac { ln(-z) }{ 2\pi i } +\frac { 1 }{ 2 } \right) /z\notin (0,1)\]

Therefore, I'll re-write \(S\) as: \[S=\frac { 1 }{ 2 } \frac { { \left( 2\pi \right) }^{ 2 } }{ m! } { B }_{ 2 }\left( \frac { x }{ 2\pi } \right) =\frac { { \pi }^{ 2 } }{ 6 } -\frac { \pi x }{ 2 } +\frac { x^{ 2 } }{ 4 } \]

Note by Aditya Kumar
8 months, 2 weeks ago

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Alternatively, show that the LHS is the Fourier Series of the RHS for \(0\leq x\leq 2\pi\). It suffices to show that \(\displaystyle \int_{0}^{2\pi}\left(\frac{\pi^2}{6}-\frac{\pi x}{2}+\frac{x^2}{4}\right)\cos(kx)dx=\frac{\pi}{k^2}\), a simple exercise in calculus.

Note that the equation holds for \(0\leq x\leq 2\pi\) only, a fact that should be stated in the problem. Otto Bretscher · 8 months, 2 weeks ago

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@Otto Bretscher Sir why did you remove your problem? Aditya Kumar · 8 months, 2 weeks ago

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@Aditya Kumar I had no idea the same problem was posed before... It can happen to all of us Otto Bretscher · 8 months, 2 weeks ago

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@Otto Bretscher Sir can you give me the link? Aditya Kumar · 8 months, 2 weeks ago

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@Aditya Kumar Somebody posted it as a comment to the problem I have deleted... I really don't remember it... it was an old problem that I must have missed Otto Bretscher · 8 months, 2 weeks ago

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@Otto Bretscher Can you at least post your method? Aditya Kumar · 8 months, 2 weeks ago

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@Aditya Kumar I'm at work now, but I can post something in the evening. Again, I was using Fourier series. Otto Bretscher · 8 months, 2 weeks ago

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@Otto Bretscher Sure sir no problem. Aditya Kumar · 8 months, 2 weeks ago

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@Aditya Kumar Maybe you can try it yourself: Write the Fourier series of \(x\) and \(x^3\) on \([\pi,\pi]\); then take the linear combination that gives you what you want. Otto Bretscher · 8 months, 2 weeks ago

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@Otto Bretscher Sir is this your book? I seriously liked it. Aditya Kumar · 8 months, 2 weeks ago

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@Aditya Kumar Yes, I must confess, it is.... the book is generally not very popular with students (but it is used at many of the best universities in the US) Otto Bretscher · 8 months, 2 weeks ago

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@Otto Bretscher I'm trying that :) Aditya Kumar · 8 months, 2 weeks ago

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@Otto Bretscher Wow. Thanks. Aditya Kumar · 8 months, 2 weeks ago

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There is a sign error: We have \(\cos(kx)=\frac{e^{ikx}+e^{-ikx}}{2}\) Otto Bretscher · 8 months, 2 weeks ago

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@Otto Bretscher Thanks for mentioning it sir. It was a typo. While doing it on paper I hadn't made that mistake. Aditya Kumar · 8 months, 2 weeks ago

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