Analytic functions - necessary and sufficient conditions for analyticity in cartesian and polar co-ordinates

ANALYTIC FUNCTIONS - NECESSARY AND SUFFICIENT CONDITIONS FOR ANALYTICITY IN CARTESIAN AND POLAR CO-ORDINATES

 (a) Analytic [or] Holomorphic [or] Regular function

[A.U A/M 2019 R-13]

A function is said to be analytic at a point if its derivative exists not only at that point but also in some neighbourhood of that point.

[or]

A function f (z) which is single-valued and possesses a unique derivative with respect to z at all points of a region R, is called an analytic or a regular function of z in that region.

Note (i) The synonyms for analytic are holomorphic and regular.

(b) Entire function: [Integral function]

A function which is analytic everywhere in the finite plane is called an entire function.

An entire function is analytic everywhere except at z = ∞.

Example: e², sin z, cos z, sinh z, cosh z

(i) The necessary condition for f (z) to be analytic  [Cauchy - Riemann Equations]

[A.U D15/J16 R-08]

The necessary conditions for a complex function f(z) = u(x, y) + iv (x, y) to be analytic in a region R are ∂u/∂x = ∂v/∂y and ∂v/∂x = - ∂u/∂y i.e., u_x = v_y and v_x = - u_y

 [OR]

Derive C R equations as necessary conditions for a function w = f (z) to be analytic. [Anna, Oct. 1997] [Anna, May 1996]

Proof : Let fz) = u(x, y) + iv (x, y) be an analytic function at the point z in a region R. Since f (z) is analytic, its derivative f' (z) exists in R

The above equations are known as Cauchy - Riemann equations or C-R equations.

Note (ii) The above conditions are not sufficient for f (z) to be analytic. The sufficient conditions are given in the next theorem.

(ii) Sufficient conditions for f (z) to be analytic.

If the partial derivatives ux, uy, V, and vy are all continuous in D and Ux = Vy and uy = -Vx, then the function f (z) is analytic in a domain D.

(iii) Polar form of CR equations

In Cartesian co-ordinates any point z is z = x + iy.

In polar co-ordinates, z = re^iθ where r is the modulus and θ is the argument.