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how many zero's ?

When trying to glue some ideas together i often stumble upon the problem of finding zero's / fixpoints and the amount of zero's / fixpoints.

Not just the computation , but the proof for such a problem with real parameters.

Despite the existence of complex analysis tools such as the argument principle , fractals , lagrange multipliers and riemann surfaces the problem still seems " hard " in the general case.

Maybe it requires just a lot of work and educated guesses and the use of rouché theorem but i would like a more systematic approach.

A good example of what i mean is this:

Let the amount of distinct complex zero's in terms of the real parameters a,b,c be written as N(a,b,c).

Express N(a,b,c) for the function f(z) - z :=

2*sinh(a*z*(1+exp(b*z^2 + c*z^4))) - z

How to deal with that ?

Maybe consider abs ( f ' (z) ) = 1 ?



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