# Dictionary:Analytic Function

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Of all of the functions of a *complex variable,* one of the most useful varieties are the *Analytic functions.*

A *complex valued function* is a function

where such that is called the *real part* and is called the *imaginary part* of .

## Definition of *Analytic function*

The term *analytic* (also known as *regular* or *holomorphic*) function is a function such that the
derivative with respect to exists.

These two equations are known as the Cauchy-Riemann equations

Any complex valued function of the complex variable
that satisfies the Cauchy-Riemann equations is called *analytic*.

Analytic functions are not rare. Most, if not all of the functions encountered applied mathematics likely satisfy
the Cauchy-Riemann equations except possibly at isolated points. These isolated points are *poles,* *branch points,*
and *essential singularities.*

### Poles

If is analytic everywhere in the complex plane, it is called *entire.* Examples of entire functions include
polynomials in just to name a few.
We typically talk about analyticity in some region of the complex plane.

We can consider a class of functions known as *meromorphic* functions, which are functions that have poles. A pole is
an algebraic singularity in the denominator, such as would be given by

in some region of the complex plane, where is analytic and is an integer.

### Branch points

A *branch point* is a failure of analyticity owing to multivaluedness of a function. For example, if

with analytic, but with (either positive or negative) but with not an integer. Thus, the multivaluedness results from the roots of the function.

Other functions that contain branch points include logarithm whose multivaluedness maybe seen by noting that implying that

Note that while branch points are well defined, the notion of a *branch cut* that is often seen in
mathematical literature is an artifact
of the choice of a reference direction, which need not be a straight line, but could be any curve (such as a path of
integration). If we attempt rotate the points on that curve
around a branch point then we become aware of the multivaluedness of the function. Care must be taken so as not to introduce
*branch cuts* unnecessarily, as these may complicate analysis.

### Essential singuarities

For the case of , the Laurent series of
the function has an infinite number of negative power terms, which are singular at a point. That point is called an
*essential singularity.*

### Other properties of analytic functions

Analytic functions have properties that make them desirable:

- If a function is analytic, then so is its derivative.

- Analytic functions are infinitely differentiable.

- The real and imaginary parts of an analytic function satisfy Laplace's equation, making them ideal for electromagnetic representations.

- The integral over a closed contour of a function that is analytic inside and on the contour is zero.

- A function that is analytic has a convergent power series in that region, which is the same as the Taylor expansion, which exists, converges, and is unique in the region of analyticity.