Website owner: James Miller
Analytic continuation
Analytic continuation. Let f1(z) be a function that is analytic in a region R1 of the complex plane and f2(z) a function that is analytic in a region R2 partly overlapping R1. If f1(z) = f2(z) in the overlapping part, then f2(z) is called the analytic continuation of f1(z). This means that there is a function f(z) that is analytic in the combined regions R1 and R2 such that f(z) = f1(z) in R1 and f(z) = f2(z) in R2. It is sufficient for R1 and R2 to have only a small arc in common such as the arc ABC shown in Fig. 2.
By analytic continuation to regions R3, R4, etc. we can extend the original region of definition to other parts of the complex plane. The functions f1(z), f2(z), f3(z), ... defined in R1, R2, R3, ... respectively, are called function elements or briefly elements. It is sometimes impossible to extend a function analytically beyond the boundary of a region. In such a case the boundary is called a natural boundary.
Uniqueness theorem for analytic continuation. Let a function f1(z) defined in R1 be continued analytically to region Rn along two different paths. See Fig. 3. Then the two analytic continuations will be identical providing there is no singularity between the paths.
If we do get different results when using two different paths we can show that there is a singularity (specifically a branch point) between the paths.
One can illustrate analytic continuation with Taylor series expansions. Suppose we do not know the exact form of an analytic function f(z) but only know that inside some circle of convergence C1 with center at a f(z) is represented by a Taylor series
1) a0 + a1(z - a) + a2(z - a)2 + ...
If we then choose a point b inside C1 we can find the value of f(z) and its derivatives at b from 1) and thus arrive at a new series
2) b0 + b1(z - b) + b2(z - b)2 + ...
having a circle of convergence C2. See Fig. 4. We can then choose a point c inside C2 and repeat the process. The process can be repeated indefinitely.
The collection of all such power series representations, i.e. all possible analytic continuations, is defined as the analytic function f(z).
References
Spiegel. Complex Variables. (Schaum)
Jesus Christ and His Teachings
Way of enlightenment, wisdom, and understanding
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
On Self-sufficient Country Living, Homesteading
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
Theory on the Formation of Character
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life