# EllipticLog

EllipticLog[{x,y},{a,b}]

gives the generalized logarithm associated with the elliptic curve .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• EllipticLog[{x,y},{a,b}] is defined as the value of the integral , where the sign of the square root is specified by giving the value of y such that .
• EllipticLog can be evaluated to arbitrary numerical precision.

# Examples

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## Basic Examples(2)

Evaluate numerically:

Plot over a subset of the reals:

## Scope(16)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

### Specific Values(3)

Value at fixed points:

Value at zero:

Find a value of for which Abs[EllipticLog[{x,Sqrt[x^3+5x^2+x]},{5,1}]]=0.8:

### Visualization(2)

Plot the EllipticLog function:

Plot the real part of EllipticLog[{z,Sqrt[z^3+2 z^2+ z]},{2,1}]]:

Plot the imaginary part of EllipticLog[{x+ y,Sqrt[z^3+2 z^2+ z]},{2,1}]]:

### Function Properties(3)

EllipticLog is not an analytic function:

It has both singularities and discontinuities: is neither non-negative nor non-positive: is neither convex nor concave:

### Differentiation(2)

First derivative with respect to :

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

## Applications(2)

Define multiplication on the elliptic curve :

Use multiplication on the elliptic curve to add rational numbers:

The value of EllipticLog at the product point equals the sum of values of EllipticLog at the corresponding factors: