Euler's series

As Euler's number is the identity

${\displaystyle {\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {\sin(2\pi nx)}{n}}={\frac {1}{2}}-x,\;0<x<1}$

designated.

Leonhard Euler communicated the Euler's series in his letter of July 4, 1744 to Christian Goldbach , but without proof. Almost ten years later he published a proof in his Institutiones calculi differentialis . Euler's series is a function that can be easily developed into a Fourier series . The Bernoulli polynomials and Poisson's sum formula can be traced back to this series, which is fundamental for analysis.

Euler's series forms the imaginary part of the series

${\displaystyle \sum _{n=1}^{\infty }{\frac {e^{2\pi inx}}{n}},\;x\in \mathbb {R} ,\;x\notin \mathbb {Z} }$

main clause

Given the interval . Furthermore, let two points out . The following series of functions converges uniformly on and the following applies: ${\displaystyle I:=(0,1)}$ ${\displaystyle a,b\in I,\ a<b}$ ${\displaystyle I}$ ${\displaystyle {\bar {I}}:=[a,b]}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {e^{2\pi inx}}{n}}=-\log \left[2\sin(\pi x)\right]+i\pi \left({\frac {1}{2}}-x\right),\;0<x<1}$

literature

• Max Koecher : Classical elementary analysis , Birkhäuser Verlag , Basel-Boston, 1987

Web links

• Elaboration on Euler's series with proof (PDF file; 131 kB)