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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: $S=\frac { 1 }{ 2 } \frac { { \left( 2\pi \right) }^{ 2 } }{ m! } { B }_{ 2 }\left( \frac { x }{ 2\pi } \right) =\frac { { \pi }^{ 2 } }{ 6 } -\frac { \pi x }{ 2 } +\frac { x^{ 2 } }{ 4 }$

1 year, 4 months 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. · 1 year, 3 months ago

Sir why did you remove your problem? · 1 year, 3 months ago

I had no idea the same problem was posed before... It can happen to all of us · 1 year, 3 months ago

Sir can you give me the link? · 1 year, 3 months ago

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 · 1 year, 3 months ago

Can you at least post your method? · 1 year, 3 months ago

I'm at work now, but I can post something in the evening. Again, I was using Fourier series. · 1 year, 3 months ago

Sure sir no problem. · 1 year, 3 months ago

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. · 1 year, 3 months ago

Sir is this your book? I seriously liked it. · 1 year, 3 months ago

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) · 1 year, 3 months ago

I'm trying that :) · 1 year, 3 months ago

Wow. Thanks. · 1 year, 3 months ago

There is a sign error: We have $$\cos(kx)=\frac{e^{ikx}+e^{-ikx}}{2}$$ · 1 year, 3 months ago