Mathematical problems
Preface. I have some hints, how to solve the problems below, and I try to write it out.
However, if you can advance faster than I do, go ahead. Questions are welcomed.
Prove the Theorem A below:
Let t be entire 1-periodic function, different from a constant, and
J(z)=z+t(z) for all complex z, and
let S be set of complex numbers with real part within segment [0,1].
Then:
For any complex number p, except, perhaps, one value,
in the domain S, there exist solution u of equation J(u)=p,
and this solution is not unique.
Prove the Theorem B below:
Let
C be set of complex numbers z such that either |Im(z)|>0 or z>-2.
Then:
There exist function F holomorphic on C such that
F(z*)=F(z)*
for all z from C and
exp(F(z))=F(z+1) for all z from C
and F(0)=1.
There exist only one such function.
Make the Construction A below:
Let
C be set of complex numbers such that Im(z)>0 or z>1.
Construct some function h holomorphic on C such that
h(z*)=h(z)* for all z from C and
h(h(z))=z! for all z from C.
Suggest a way for evaluation of such a function.
Make the Construction B below:
For a given d>0, let B be set of complex numbers z such that
at least one of the two conditions holds:
|z| < 1 or { Re(z)>0 , |Im(z)|<d } .
Let C be set of all complex numbers that do not belong to B.
Construct some entire function F, such that
F(0)=1 and |F(z)|<|z| for all z in C.
Suggest a way for evaluation of such a function.
Conclusions
Other problems require more formulas; it is difficult to type them in html.
I shall post them as pdf.
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