Website owner:  James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

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).


  Spiegel. Complex Variables. (Schaum)

More from

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

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

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together


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

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

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

Sizing up people

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

We are our habits

What creates moral character?

[ Home ] [ Up ] [ Info ] [ Mail ]