# The Modular Form Math Online Tom Circle

Form” : Function with special properties – eg.

• Space Forms: manifolds with certain shape.
• Quadratic Forms (of weight 2): \$latex x^2+3xy+7z^2 \$
• Cubic Forms (of weight 3): \$latex x^3+{x^2}y + y^3 \$
• AutomorphicForms (particular case: ModularForms): auto (self), morphic (shape).

1. Non-Euclidean Geometry

1.1 Hyperbolic Plane : is the Upper-Half in Complex plane H (positive imaginary part) where :

• Through point p there are 2 lines L1 & L2 (called “geodesic“) parallel to line L.
• Distance between p & q in H: \$latex boxed {int_{L} frac {ds}{y}}&fg=aa0000&s=2\$
where L the “line” segment (the arc of the semicircle or the vertical segment) and \$latex ds^2 = dx^2+dy^2\$ 1.2 Group of Non-Euclidean Motions:
\$latex f: H rightarrow H\$

1. Translation: \$latex z rightarrow {z + b} quad forall b in mathbb {R}\$
2. Dilation: \$latex z rightarrow {az } quad forall a in mathbb {R^{+}}\$
3. Inversion: \$latex…

View original post 741 more words ## Author: tomcircle

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