Variants of the Clausen Functions

To Prove: k=1cos(kx)k2=π26πx2+x24\sum _{ k=1 }^{ \infty }{ \frac { \cos { \left( kx \right) } }{ { k }^{ 2 } } } =\frac { { \pi }^{ 2 } }{ 6 } -\frac { \pi x }{ 2 } +\frac { x^{ 2 } }{ 4 }

0x2π0\leq x\leq 2\pi

Proof:

Write cos(kx)\cos(kx) as: cos(kx)=eikx+eikx2\displaystyle \cos { \left( kx \right) } =\frac { { e }^{ ikx }+{ e }^{ -ikx } }{ 2 }

Then we get S=k=1eikx+eikx2k2S=\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=Li2(eikx)+Li2(eikx)2S=\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: Lim(z)=(1)m1Lim(1z)(2πi)mm!Bm(ln(z)2πi+12)/z(0,1){ 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 SS as: S=12(2π)2m!B2(x2π)=π26πx2+x24S=\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
3 years, 7 months ago

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Alternatively, show that the LHS is the Fourier Series of the RHS for 0x2π0\leq x\leq 2\pi. It suffices to show that 02π(π26πx2+x24)cos(kx)dx=πk2\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 0x2π0\leq x\leq 2\pi only, a fact that should be stated in the problem.

Otto Bretscher - 3 years, 7 months ago

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Wow. Thanks.

Aditya Kumar - 3 years, 7 months ago

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Sir why did you remove your problem?

Aditya Kumar - 3 years, 7 months ago

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I had no idea the same problem was posed before... It can happen to all of us

Otto Bretscher - 3 years, 7 months ago

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@Otto Bretscher Sir can you give me the link?

Aditya Kumar - 3 years, 7 months 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 - 3 years, 7 months ago

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@Otto Bretscher Can you at least post your method?

Aditya Kumar - 3 years, 7 months 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 - 3 years, 7 months ago

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@Otto Bretscher Sure sir no problem.

Aditya Kumar - 3 years, 7 months ago

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@Aditya Kumar Maybe you can try it yourself: Write the Fourier series of xx and x3x^3 on [π,π][\pi,\pi]; then take the linear combination that gives you what you want.

Otto Bretscher - 3 years, 7 months ago

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@Otto Bretscher I'm trying that :)

Aditya Kumar - 3 years, 7 months ago

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@Otto Bretscher Sir is this your book? I seriously liked it.

Aditya Kumar - 3 years, 7 months 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 - 3 years, 7 months ago

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There is a sign error: We have cos(kx)=eikx+eikx2\cos(kx)=\frac{e^{ikx}+e^{-ikx}}{2}

Otto Bretscher - 3 years, 7 months ago

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Thanks for mentioning it sir. It was a typo. While doing it on paper I hadn't made that mistake.

Aditya Kumar - 3 years, 7 months ago

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