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stevendaryl
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zeta.png

Euler’s amazing identity
The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous equation is ##e^{i \pi} = -1##, which is one of the most important equations in modern mathematics, but unfortunately, it wasn’t invented by Euler.Something that is original with Euler is this amazing identity:
Equation 1: ##1 + 1/4 + 1/9 + 1/16 + … = \pi^2/6##
This is one instance of an important function called the Riemann Zeta function, ##\zeta(s)##, which in the case where ##s > 1## is defined by:
Equation 2: ##\zeta(s) = \sum_{j=1}^\infty \dfrac{1}{j^s}##
So Euler’s identity can be written as:
Equation 3: ##\zeta(2) = \frac{\pi^2}{6}##
This post is an attempt to show how you can derive that result, and related results, using facts about trigonometry, complex numbers, and the Fourier series.
Some related functions defined...

Continue reading...
 

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  • #2
SSequence
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Just below the heading "Equation-7". Also same identity below the heading "Equation-9" and before "Sum-1".
##F(x)= \sum_{j=-\infty}^{\infty} e^{ijx}=\sum_{j=-\infty}^{-1} e^{ijx}+1+\sum_{j=0}^{\infty} e^{ijx} ##

Shouldn't it be(?):
##F(x)= \sum_{j=-\infty}^{\infty} e^{ijx}=\sum_{j=-\infty}^{-1} e^{ijx}+1+\sum_{j=1}^{\infty} e^{ijx} ##
 
  • #6
Svein
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I've checked chrome and safari and your link is broken in both. There appears to be garbage prior to the working URL. Is this the correct one?
Yes. I have extended the results in that insight and I will update it Really Soon Now (as Jerry Pournelle used to say).
 

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