Problem with cauchy method ?

[tommy1729]

The Cauchy method has the property/condition (assum it works) :

exp(f(-1 +ai)) = f(ai)
exp(f(ai)) = f(1+ai)

for real a.

But is that sufficient to conclude f(z+1) = exp(f(z)) ?
Compare to x sin(x/2 pi) for integer x.

However the answer is YES ITS SUFFICIENT.
Because exp(f(-1 +ai)) - f(ai) = 0 and then use analytic continuation.

But what about the theta wave ? How do we know we arrive at a (unique) solution that is bounded in the strip ?

regards

tommy1729