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

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
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Comments

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 - 5 years, 2 months ago

Wow. Thanks.

Aditya Kumar - 5 years, 2 months ago

Sir why did you remove your problem?

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

@Otto Bretscher – Sir can you give me the link?

@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 – Can you at least post your method?

@Aditya Kumar – I'm at work now, but I can post something in the evening. Again, I was using Fourier series.

@Otto Bretscher – Sure sir no problem.

@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 – I'm trying that :)

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

@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)

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

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

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Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$