Given:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] f\left(\frac{az + b}{cz + d}\right) = (cz + d)^kf(z)[/tex]

We can apply:

[tex]\left( \begin{array}{cc}

a & b \\

c & d\\

\end{array} \right)

= \left( \begin{array}{cc}

1 & 1 \\

0 & 1 \\

\end{array} \right)[/tex]

So that we arrive at the periodicity [itex] f(z+1) = f(z) [/itex]. This implies a Fourier expansion:

[tex]f(z) = \sum_{n=0}^{\infty}c_nq^n[/tex]

Where [itex]q = e^{2{\pi}inz}[/itex]

But how to calculate the coefficients [itex]c_n[/itex]? Scouring all over the internet, I've seen mention of contour integration and parametrizing along the Half-Plane, but I'm not even sure of the form of the integrand. Ideas would be most appreciated.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Coefficients of Modular Forms

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - Coefficients Modular Forms | Date |
---|---|

I Help with expression ##F(it)-F(-it)## in the Abel-Plana form | Aug 7, 2017 |

I Does analysis form a bridge to geometry? | Dec 21, 2016 |

A Taking limits in discrete form | Mar 14, 2016 |

Identical Fourier coefficients of continuous ##f,\varphi\Rightarrow f=\varphi## | Nov 17, 2014 |

Fourier series: relation of coefficients | Jan 12, 2013 |

**Physics Forums - The Fusion of Science and Community**