In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, given below.
See main article: Exponential type. A function
f(z)
M
\alpha
|f(rei\theta)|\leMe\alpha
in the limit of
r\toinfty
z
z=rei\theta
\theta
\alpha
\alpha
f
\alpha
For example, let
f(z)=\sin(\piz)
\sin(\piz)
\pi
\pi
\sin(\piz)
\pi
Bounding may be defined for other functions besides the exponential function. In general, a function
\Psi(t)
infty | |
\Psi(t)=\sum | |
n=0 |
\Psintn
with
\Psin>0
n
\limn\toinfty
\Psin+1 | |
\Psin |
=0.
Comparison functions are necessarily entire, which follows from the ratio test. If
\Psi(t)
f
\Psi
M
\tau
\left|f\left(rei\theta\right)\right|\leM\Psi(\taur)
as
r\toinfty
\tau
\tau
f
\Psi
\tau
Nachbin's theorem states that a function
f(z)
infty | |
f(z)=\sum | |
n=0 |
fnzn
is of
\Psi
\tau
\limsupn\toinfty\left|
fn | |
\Psin |
\right|1/n=\tau.
Nachbin's theorem has immediate applications in Cauchy theorem-like situations, and for integral transforms. For example, the generalized Borel transform is given by
infty | |
F(w)=\sum | |
n=0 |
fn | |
\Psinwn+1 |
.
If
f
\Psi
\tau
F(w)
|w|\le\tau.
Furthermore, one has
f(z)= | 1 |
2\pii |
\oint\gamma\Psi(zw)F(w)dw
where the contour of integration γ encircles the disk
|w|\le\tau
\Psi(t)=et
\alpha(t)
[0,infty)
1 | |
\Psin |
=
infty | |
\int | |
0 |
tnd\alpha(t)
where
d\alpha(t)=\alpha\prime(t)dt
F(w)= | 1 |
w |
infty | |
\int | |
0 |
f\left(
t | |
w |
\right)d\alpha(t).
The ordinary Borel transform is regained by setting
\alpha(t)=e-t
Nachbin resummation (generalized Borel transform) can be used to sum divergent series that escape to the usual Borel summation or even to solve (asymptotically) integral equations of the form:
infty | |
g(s)=s\int | |
0 |
K(st)f(t)dt
where
f(t)
K(u)
f(x)=
infty | |
\sum | |
n=0 |
an | |
M(n+1) |
xn
g(s)=
infty | |
\sum | |
n=0 |
ans-n
M(n)
K(u)
\pi(x) ≈
infty | |
1+\sum | |
n=1 |
logn(x) | |
n ⋅ n!\zeta(n+1) |
.
in some cases as an extra condition we require
infty | |
\int | |
0 |
K(t)tndt
n=0,1,2,3,...
Collections of functions of exponential type
\tau
\|f\|n=\supz\exp\left[-\left(\tau+
1 | |
n |
\right)|z|\right]|f(z)|.