As we move away from the direct pages of my books, yet carry
forward the insights and techniques developed within them, Episode 6 marks a
new stage in our exploration. Here, we leverage the foundational knowledge from
The Essential Transform Toolkit and The Riemann Hypothesis Revealed
to construct more mathematical results and perspectives. This episode is
dedicated to showing how the core ideas—analyticity, contour integration, and
the interplay between power series and entire functions—can be extended and
applied beyond the original context. Our journey now is not just about
revisiting established results, but about building upon them, discovering how
the methods and intuition gained from previous episodes empower us to tackle
broader questions and uncover deeper connections in complex analysis and beyond.
Introduction
The function
is analytic for 
. Prove that the series
converges in the whole plane and the estimates
(
is a constant) hold for
its sum.
We are given a power series 
which is analytic for . This implies the series converges uniformly on , and in particular is bounded there.
Cauchy estimates for the coefficients
Since 
is analytic on , by the maximum modulus principle (see The Riemann Hypothesis
Revealed for a comprehensive treatment of the subject) there exists a constant 
such that
By Cauchy’s integral formula for
coefficients, we have
hence
Thus,
Convergence of 
in the whole plane
Consider
Using the estimate above,
The series
is exactly 
, which converges for all 
.
Hence, by comparison, 
converges absolutely for
all , so 
is an entire function.
Growth estimate for 
Summing the inequality termwise,
Thus,
Estimates for the derivatives
Differentiate termwise:
Using the coefficient estimate again,
Summing over 
,
Therefore,
This result is a classical bridge between analyticity on a
disk and entire functions of exponential type. It is closely related
to Laplace–Borel transforms.
The idea behind Laplace-Borel transforms is to start with a
(possibly divergent) power series
then define its Borel transform to be the exponential series
The Borel transform always converges as a formal power series if
has a positive radius of
convergence. What really matters is not mere convergence, but analytic
continuation and controlled growth of 
.
Thus, the idea is to replace a power series whose coefficients
grow like 
by a new series whose
coefficients grow moderately, so that it defines an analytic function with at most exponential
growth. This function can then be resummed by a Laplace-type integral to
reconstruct (or analytically continue) .
I’ll show two things:
For 
and 
(i.e., is analytic for 
and
The function 
provides the analytic
continuation of 
into a domain 
defined as follows:
Imagine a straight line is drawn through
every singularity of perpendicular to the
segment connecting this singularity to the origin. The domain defined as the largest convex region containing
the disk , and its boundary is made up of the points where these
perpendicular lines are drawn.
If there are only a finite number of
singularities, the boundary of will be a polygon.
From the previous result we know that is entire and satisfies
The Laplace–Borel identity for 
Consider for ,
Integrating by parts we get
From the estimate
it follows that
and
since .
Thus,
Conclusion
For ,
This is the Borel-Laplace inversion formula.
Analytic continuation via the Laplace integral
I’ll show that
defines an analytic continuation of to a larger domain .
Where does the integral converge?
Suppose that has a singularity at a
point 
with 
.
Draw the line perpendicular to the segment 
, passing through . This line satisfies
When 
lies on it, the Laplace
integral becomes obstructed. Particularly, for 
, the growth of 
wins 
and the integral
diverges. Thus, the singularity at erects a hard barrier
along the perpendicular line.
The domain 
Repeat this for every singularity of on 
and intersect all
corresponding half-planes defined by 
. The intersection forms the convex domain . If there are only finitely many singularities, this domain
will be a polygon.
Analyticity of 
To see why 
is analytic, pick such that and draw a circle 
of diameter 
and a circle 
, concentric to of radius 
for sufficiently small 
.
There are no singularities of the function within and on the
boundary of . The equality
still holds for the coefficients 
of the expansion
and consequently
Since the series
converges uniformly on
stays away from 
, hence there exists 
such that 
is bounded on
hence from Weierstrass M-Test
it follows that
The maximum of 
on 
is 
.
Indeed, the quantity 
is invariant under
simultaneous rotation of and . Thus, we may assume without loss of generality that 
real and positive. It
follows that
Now write
So
For non-zero ,
On the circle : 
is maximal when 
and 
is minimal at the same
point. That point is
Thus,
Therefore,
This gives
which ensures the exponential decay in the Laplace integral and
justifies analyticity.
Why this proves analyticity of 
:
From
we see:
for each fixed 
, 
is entire in ;
the dependence of is through an exponential
kernel;
the integral defining
is uniformly dominated on compact subsets
of .
Final conclusion
The identity
holds.
Moreover, this same integral defines an analytic function beyond
. In fact, it provides an analytic continuation of into the convex domain bounded by the
perpendicular lines to the rays through the singularities of .
This is a classical result in Borel summation theory and
underlines the connection between power series, exponential type entire
functions, and convex analytic continuation domains.
Epilogue
In this episode, we have seen how the
knowledge and techniques distilled from my books serve as a springboard for
further discovery. By moving beyond the confines of the original texts, we have
constructed new results and gained a richer understanding of analytic
continuation, entire functions, and the power of Laplace–Borel transforms. This
process of building—layer by layer—demonstrates the enduring value of a solid
foundation: once the core principles are mastered, they become versatile tools
for innovation and exploration. As we continue to move forward, let us remember
that the true strength of mathematical knowledge lies not only in what we learn
from the classics, but in how we use that knowledge to create, connect, and
push the boundaries of what is possible.
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