References requested.
Anderson Brasil
another entire function f(z) such that f(z+1)-f(z)=g(z) for every z in
the complex domain.
Has someone ever studied that problem? Is it related to any part of
mathematics?
Any reference would be appreciated.
Thanks in advance,
(and apologizing for this cold murder of the english language)
Anderson Brasil
ander...@alternex.com.br
Chip Eastham
Anderson,
If only my Portuguese could "hold a candle" to your English!
Just a thought, but try to show the problem soluble for any
polynomial/monomial g(z), and apply the results to a power
series expansion of g(z) about the origin. You say that g
is entire, so the radius of convergence would be infinite.
Regards,
Chip
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Torsten Ekedahl
> Consider the following problem. Given an entire function g(z), find
> another entire function f(z) such that f(z+1)-f(z)=g(z) for every z in
> the complex domain.
>
> Has someone ever studied that problem? Is it related to any part of
> mathematics?
>
The relation I can think of is the following:
Suppose one wants to computer the cohomology of the structure sheaf of
the complex manifold C^*. Of course, this is a Stein manifold so all
higher cohomology vanishes. On the other hand, we may use the regular
covering space map exp: C --> C^* and use the Hochschild-Serre
spectral sequence to compute. The group of deck transformations is Z
so that the E_2 term of it is H^i(Z,H^j(C,O)), where O is the
structure sheaf. As C also is Stein H^j(C,O)=0 when j > 0 and so we
get an equality
H^i(Z,E) = H^i(C^*,O),
where E is the space of entire functions. Now, for any Z-module E
H^1(Z,E) is the cokernel of the map e |--> se - e, where s is the
generator of Z (1 that is). Hence H^1(C^*,E) is seen as the
obstruction for solving the equation f(z+1)-f(z)=g(z), i. e., the
equation is solvable iff the image of g in it is zero. Now, as C^* is
Stein H^1(C^*,O)=0 so the equation is always solvable.
Michel Balazard
Look in Berenstein-Gay, Special topics in complex and harmonic analysis.
Vasyl Yaremenko
> ander...@alternex.com.br (Anderson Brasil) wrote:
> Consider the following problem. Given an entire function g(z), find
> another entire function f(z) such that f(z+1)-f(z)=g(z) for every z in
> the complex domain.
>
> Has someone ever studied that problem? Is it related to any part of
> mathematics?
> Any reference would be appreciated.
> Thanks in advance,
> (and apologizing for this cold murder of the english language)
> Anderson Brasil
> ander...@alternex.com.br
>Anderson,
I have studied the problem which is analogous to your problem but
concerning with real functions depending on integer argument i.
The function g(i) = f(i+1) - f(i) I consider as
discrete derivative of the function f(i) which is a discrete prototypal
of the function g(i). I have stated that in the case when g(i) is
a polynomial the correspondent prototypal f(i) is also a polynomial
and may be determined by solution of some set of simultaneous
linear equations.
I have also discovered that on the base of the definition of a
discrete derivative the discrete analysis may be built which is in
many respects like the known Newton-Leibnitz's (continuous) analysis. The
discrete analysis allows to derive formulas for
series sums (including composite sums) like to deriving formulas for squares
and volumes by the continuous analysis. My main result
is the proof of the _exact_ formula like to the Tailor's formula.
It is also interesting that the polynomial used for definition of a
Bernoulli's number is the prototypal of the correspondent power
function.
Unfortunately, I don't know if my results are new. If you are
interesting in checking possibility to generalize my results
on the complex domain I could sent my article to you.
Best wishes,
Vasyl Yaremenko