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gives the Klein invariant modular elliptic function .
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The argument is the ratio of Weierstrass half-periods .
  • is invariant under any combination of the modular transformations and .
  • For certain special arguments, KleinInvariantJ automatically evaluates to exact values.
Evaluate numerically:
Evaluate numerically:
Click for copyable input
Click for copyable input
Evaluate to high precision:
The precision of the output tracks the precision of the input:
KleinInvariantJ threads element-wise over lists:
TraditionalForm formatting:
Some modular properties of KleinInvariantJ are automatically applied:
Verify a more complicated identity numerically:
Find values at quadratic irrationals:
KleinInvariantJ is a modular function. Make an ansatz for a modular equation:
Form an overdetermined system of equations and solve it:
This is the modular equation of order 2:
Solution of the Chazy equation :
Plot the solution:
Plot the absolute value in the complex plane:
Plot the imaginary part in the complex plane:
Find derivatives:
Find a numerical root:
Machine-precision input is insufficient to give a correct answer:
With exact input, the answer is correct:
KleinInvariantJ remains unevaluated outside of its domain of analyticity:
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