Formelsammlung Mathematik: Unendliche Reihen: Reihen mit Integralkosinus

Zurück zu Unendliche Reihen

1

$2\cdot \sum _{n=1}^{\mathrm{\infty }}\text{Ci}\left(\frac{n\pi }{2}\right)=2\log 2-\gamma$

2

$2\cdot \sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\text{Ci}\left(\frac{n\pi }{2}\right)=-\gamma$

3

$2\cdot \sum _{n=1}^{\mathrm{\infty }}\text{Ci}(n\pi )=\log 2-\gamma$

4

$2\cdot \sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\text{Ci}(n\pi )=1-\log 2-\gamma$

5

$2\cdot \sum _{n=0}^{\mathrm{\infty }}\text{Ci}\left((n+\frac{1}{2})\pi \right)=\log 2$

6

$2\cdot \sum _{n=1}^{\mathrm{\infty }}\text{Ci}(2n\pi )=\frac{1}{2}-\gamma$

7

$2\cdot \sum _{n=1}^{\mathrm{\infty }}\text{Ci}((2n+1)\pi )=\log 2-\frac{1}{2}$

8

$2\cdot \sum _{n=1}^{\mathrm{\infty }}\text{Ci}(n\pi x)=\psi (1+\lfloor \frac{x}{2}\rfloor )-\log \frac{x}{2}$

9

$2\cdot \sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\text{Ci}(n\pi x)=\psi (\frac{1}{2}+\lfloor \frac{x+1}{2}\rfloor )-\log \frac{x}{2}$

10

$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k\pi }(\frac{\pi }{2}-\text{Si}(2\pi kx))=x-(\lfloor x\rfloor +\frac{1}{2})\log x+\log (\lfloor x\rfloor !)-\log \sqrt{2\pi }$

11

$\frac{4}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{2k+1}\text{Ci}((2k+1)\pi x)=(-1{)}^{\lfloor x+\frac{1}{2}\rfloor }\log (\pi x)-2\log \left(\sqrt{2\pi }\frac{\mathrm{\Gamma }\left(\frac{3}{4}\right)}{\mathrm{\Gamma }\left(\frac{1}{4}\right)}\right)+2\sum _{n=0}^{\lfloor x-\frac{1}{2}\rfloor }(-1{)}^n\log ((2n+1)\frac{\pi }{2})$