University Mathematics Handbook (2015)

XII. Complex Functions

Chapter 4. Complex Function Derivative

4.1  Definition

a.  If the single-valued function has a finite limit , then is said to be differentiable on .

b.  If function is differentiable on , it is continuous on that point.

4.2  Derivative Rules

If is constant, and functions and are differentiable, then:

a.  

b.  

c.  

d.  

e.  ,

f.  A composite function derivative: If is differentiable on , and is differentiable on , then composite function is differentiable on and .

g.  

h.  

i.  

j.  

4.3  Cauchy-Riemann (CR) Criterion (Equations)

a.  Function is differentiable at point if and only if and are differentiable at , and

In this case:

b.  CR criterion in polar coordinates: if

then, is differentiable if, and only if,

and its derivative is

4.4  Analytic Function

a.  Single-valued function is analytic function on if there exists neighborhood , such that is differentiable on every point of it.

b.  If a function is analytic at a point, then it is continuous on it.

c.  Analytic function is sometimes called a holomorphic function or regular function.

d.  An analytic function at every point of complex plane is called an entire function.

e.  A single-valued function is analytic on domain if it is differentiable at every point of the domain.

f.  A set of points where is analytic has to be an open set. If is said to be analytic in closed set , it means there exists open set containing , where is analytic.

g.  If functions and are analytic on , then functions , and when does not zero at any point of , are also analytic on .

h.  The composition of analytic functions is an analytic function.

i.  A single-valued branch of a multivalued function is an analytic function in domain if it is differentiable at any point of .

4.5  Harmonic Functions

a.  A two-variable real function is harmonic in domain if it is continuous and has continuous partial derivatives up to second order and satisfy the Laplace equations (see XIV.7).

b.  If function is analytic in domain , then functions and are harmonic
on .

c.  If and are harmonic on and hold the C.R. criteria on , then function is called a harmonic conjugate function to on .

d.  Function is analytic on if and only if is a harmonic conjugate to on .

e.  For every function harmonic on , there exists harmonic function conjugate to .

4.6  Conformal Mapping

a.  The transforming of one complex plane on another one is called conformal mapping or transformation at if it preserves the magnitude and direction of the angles and expands constantly in all directions. In other words, a transformation is conformal if it transforms a small enough triangle the vertex of which is on to a small enough triangle similar to it.

b.   is conformal mapping in domain if, and only if, function is analytic on and , .

c.  Riemann's theorem: There exists analytic function mapping simply connected domain on simply connected domain except in two cases:

and/or are full complex plane;

and/or are all full complex plane pierced in one point.

d.  If function is analytic in simply connected domain and continuous on , objectively mapping the boundary of on curve in plane and preserves the direction of , then conformly maps domain to a domain bounded by curve .

e.  Examples:

1.  Displacement:

2.  Rotation: ( real constant). In this case and , that is, point transforms to point by a rotation of vector around the origin at angle .

3.  Extension: . In this case, and . Therefore, point transforms to point on line at a times the distance . This mapping is expansion when , or contraction, when .

4.7  Möbius Transformation

a.  The transformation

when are constant complex numbers is called Möbius transformation or bilinear transformation.

b.  Circles and straight lines in the complex plane are called generalized circles.

c.  Möbius transformation maps generalized circles in plane to generalized circles in plane .

d.  If and are points symmetrical about circle in plane , that is, they are on the ray originating from center of circle , and the product of distances equals to the square of the radius of the circle, , then, after a Mobius mapping, their images and will be symmetrical with respect to circle, the image of circle .

e.  There exists a unique Mobius transformation mapping three different points , in plane to three different points in plane . This transformation is:

If one of the points equals , then all the differences including that point are replaced with .