Applied mathematics question with solution

 Note remember page numbers

To separate the real and imaginary parts of log z, we put x = r cos q; y = r sin q

w = log z = log (x + iy) 

⇒ u + iv = log (r cos q + ir sin q) = log r(cos q + i sin q) = loge 

ANALYTIC FUNCTION

A function f (z) is said to be analytic at a point z0

, if f is differentiable not only at z0

 but at 

every point of some neighbourhood of z0

.

A function f (z) is analytic in a domain if it is analytic at every point of the domain.

The point at which the function is not differentiable is called a singular point of the function.

An analytic function is also known as “holomorphic”, “regular”, “monogenic”.

Entire Function. A function which is analytic everywhere (for all z in the complex plane) 

is known as an entire function.

For Example 1. Polynomials rational functions are entire.

 2. | | z 2 is differentiable only at z = 0. So it is no where analytic.

Note: (i) An entire is always analytic, differentiable and continuous function. But converse 

is not true.

 (ii) Analytic function is always differentiable and continuous. But converse is not 

true.

(iii) A differentiable function is always continuous. But converse is not true

7.10 THE NECESSARY CONDITION FOR F (Z) TO BE ANALYTIC

Theorem. The necessary conditions for a function f (z) = u + iv to be analytic at all the 

points in a region R are

(i) 

y (ii) ∂

∂  , exist.

Proof: Let f (z) be an analytic function in a region R,

 f (z) = u + iv,

where u and v are the functions of x and y.

Let du and dv be the increments of u and v respectively corresponding to increments dx and 

dy of x and y.

\ f (z + dz) = (u + du) + i(v + dv)

Now f z z f z

since dz can approach zero along any path.

(a) Along real axis (x-axis)

 z = x + iy but on x-axis, y = 0

\ z = x, dz = dx, dy = 0

Putting these values in (1), we have

 ′