Dictionary:Analytic Function

{{#category_index:A|analytic function}}

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

$f(z)=u(x,y)+iv(x,y)$

where $z=x+iy$ such that $u(x,y)$ is called the real part and $v(x,y)$ is called the imaginary part of $f(z)$.

Definition of Analytic function

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

These two equations are known as the Cauchy-Riemann equations

${\frac {\partial u(x,y)}{\partial x}}={\frac {\partial v(x,y)}{\partial y}}$

${\frac {\partial v(x,y)}{\partial x}}=-{\frac {\partial u(x,y)}{\partial y}}.$

Any complex valued function $f(z)=u(x,y)+iv(x,y)$ of the complex variable $z=x+iy$ 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 $g(z)$ is analytic everywhere in the complex plane, it is called entire. Examples of entire functions include polynomials in $z,$ $\cos(z),$ $\sin(z),$ $\exp(z)$ 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

$f(z)={\frac {g(z)}{(z-a)^{n}}}$

in some region ${\mathcal {R}}$ of the complex plane, where $g(z)$ is analytic and $n\geq 1$ is an integer.

Branch points

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

$f(z)=g(z)(z-a)^{k}$

with $g(z)$ analytic, but with $k\neq 0$ (either positive or negative) but with $k$ not an integer. Thus, the multivaluedness results from the roots of the function.

Other functions that contain branch points include logarithm $\ln(z)$ whose multivaluedness maybe seen by noting that $z=|z|e^{i\phi }$ implying that

$\ln(z)=\ln(|z|e^{i\phi })=\ln |z|+i\phi .$

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 $2\pi$ 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 $e^{1/z},$ $\sin(1/z)$, $\cos(1/z),$ 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.