# Getting values of EllipticK with arguments that are very near 1

I need to evaluate EllipticK[m] very close to 1. However, when I get too close to 1 the function defaults to the exact solution for 1 , which is ComplexInfinity. I can enter a comparable number in Wolfram|Alpha and get both the exact and the numerical approximation, but nothing I have tried gets me that approximate number in Mathematica:

Wolfram|Alpha:

EllipticK[0.9999999999999999999999999999999999999999999999]

54.3458...


Mathematica:

EllipticK[0.9999999999999999999999999999999999999999999999]

ComplexInfinity

EllipticK[0.9999999999999999999999999999999999999999999999]//N

ComplexInfinity


I've tried various other combinations of precision manipulation as well, so far nothing has worked. Does anyone know how to override the exact answer?

### Edit

I discovered half the solution immediately after posting. When I put a 0 at the end of the string of 9s, it will evaluate the decimal approximation. However, I don't know how to implement this for the purpose of, say, plotting the EllipticK.

• Try EllipticK[0.9999999999999999999999999999999999999950] Apr 16 '19 at 23:43
• Somos means EllipticK[0.9999999999999999999999999999999999999950]. The editing is wonky with the backticks, and I don't remember how to escape them. Apr 17 '19 at 0:05
• @march Use double back-ticks: EllipticK[0.9999999999999999999999999999999999999950] Apr 17 '19 at 1:12

Your choice to evaluate EllipticK[m] for values of m near the logarithmic singularity at $$1$$ is doomed from the start. The numerically proper way to go about this is to use the relationship of the complete elliptic integral of the first kind with the arithmetic-geometric mean, so that you are evaluating an argument near $$0$$:
N[Pi/(2 ArithmeticGeometricMean[1, Sqrt[1*^-46]]), 20]
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