RamanujanTauTheta

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

τθ(z) is an analytical function of z which is defined over the whole complex z-plane.

Symmetries and periodicities

Parity

τθ(z) is an odd function.

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function τθ(z) does not have poles and essential singularities.

Branch points

Branch cuts

Series representations

Generalized power series

Expansions on branch cuts

Expansions at z==0

Expansions at z==6i

Expansions at z==-6i

Expansions at z==(Subscript[z, 0])/; (Subscript[z, 0])≠±6i±i n

Expansions at z==6i+i n

Expansions at z==-6i-i n

Asymptotic series expansions

Other series representations

Integral representations

On the real axis

Of the direct function

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Multiple arguments

Argument involving numeric multiples of variable

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Complex characteristics

Real part

Imaginary part

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

For the direct function itself

Representations through more general functions

Through other functions

Representations through equivalent functions

With related functions

History

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