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Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e)) - Printable Version

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RE: Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e)) - Ivars - 05/09/2008

Hi experts,

Please tell me what are the exact values of selfzeros on the outer shape for b= 3/2, 2, sqrt(3) , 3, 4? (it is +-I for b= e^(pi/2))?

Thank You in advance,

Ivars


RE: Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e)) - Gottfried - 03/03/2011

While the solution of the problem to find complex fixpoints t for real bases b>exp(exp(-1)) using Lambert-W which were given here satisfies all needs the problem was only unsatisfactory solved for the case of numerical evaluation when the Lambert-W including the choice of branches is not available in a software (like Pari/GP).

My own proposal was based on a binary search using a formula depending on a single parameter beta. But because the number of iterations for good approximations increase quickly I tried to improve that method by employing Newton/Raphson instead of binary search.

That led to a solution where I need only 10 to 15 iterations (number of correct digits seem to increase quadratically), and also gave an interesting powerseries for the involved function which must iterated.
Here is the link to my treatize in the discussion platform math.stackexchange:
http://math.stackexchange.com/questions/24396/how-to-find-a-newton-like-approximation-for-that-function

Gottfried