Unit 2
Complex Variables
In the narrow sense of the term, the theory of function of a complex variable is the theory of analytic functions (cf. Analytic function) of one or several complex variables. As an independent discipline, the theory of functions of a complex variable took shape in about the middle of the 19th century as the theory of analytic functions.
Complex function
x + iy is a complex variable which is denoted by z If for each value of the complex variable z = x + iy in a region R, we have one or more than one values of w = u + iv, then w is called a complex function of z. And it is denoted as w = u(x , y) + iv(x , y) = f(z) 
Neighbourhood of
Let a point in the complex plane and z be any positive number, then the set of points z such that <ε Is called ε neighbourhood of 
Limit of a function of a complex variable
Suppose f(z) is a single valued function defined at all points in some neighbourhood of point 
The
Example1: Find
Sol.
Here we have Divide numerator and denominator by , we get

Continuity A function w = f(z) is said to be continuous at z = , if
Also if w = f(z) = u(x , y) + iv(x , y) is continuous at z = then u(x , y), v(x , y) are also continuous at z = .
Differentiability
Let f(z) be a single valued function of the variable z, then
Provided that the limit exists and has the same value for all the different ways in which approaches to zero.
Example2: if f(z) is a complex function given below, then discuss

Sol.
If z→0 along radius vector y = mx But along , In different paths we get different value of that means 0 and –i/2, in that case the function is not differentiable at z = 0. 
Key takeaways
<ε 2. Limit of a function of a complex variable Suppose f(z) is a single valued function defined at all points in some neighbourhood of point  The 
A function is said to be analytic at a point if f is differentiable not only at but every point of some neighborhood at .
Note
1. A point at which the function is not differentiable is called singular point.
2. A function which is analytic everywhere is called an entire function.
3. An entire function is always analytic, differentiable and continuous function. (converse is not true)
4. Analytic function is always differentiable and continuous but converse is not true.
5. A differentiable function is always continuous but converse is not true.
The necessary condition for f(z) to be analytic
f(z) = u + i(v) is to be analytic at all the points in a region R are 1. …………. (1) 2. ……...…. (2) Provided exists Equation (1) and (2) are known as CauchyRiemann equations. 
The sufficient condition for f(z) to be analytic
f(z) = u + i(v) is to be analytic at all the points in a region R are 1. …………. (1) 2. ……...…. (2) are continuous function of x and y in region R. 
Important note
1. If a function is analytic in a domain D, then u and v will satisfy CauchyRiemann conditions.
2. CR conditions are necessary but not sufficient for analytic function.
3. CR conditions are sufficient if the partial derivative are continuous.
State and prove sufficient condition for analytic functions
Statement – The sufficient condition for a function to be analytic at all points in a region R are
1 2 are continuous function of x and y in region R. 
Proof:
Let f(z) be a simple valued function having at each point in the region R. Then CauchyReimann equation are satisfied by Taylor’s Theorem Ignoring the terms of second power and higher power We know CR equation Replacing Respectively in (1) we get
Show that is analytic at Ans: The function f(z) is analytic at if the function is analytic at z=0 Since Now is differentiable at z=0 and at all points in its neighbourhood Hence the function is analytic at z=0 and in turn f(z) is analytic at 
Example1: If w = log z, then find . Also determine where w is nonanalytic.
Sol.
Here we have Therefore and Again Hence the CR conditions are satisfied also the partial derivatives are continuous except at (0 , 0). So that w is analytic everywhere but not at z = 0 
Example2: Prove that the function is an analytical function.
Sol.
Let =u+iv Let =u and =v Hence CREquation satisfied. 
Example3: Prove that
Sol.
Given that Since V=2xy Now But Hence Given that Since V=2xy Now But Hence

Example4: Show that polar form of CR equations are
Sol
. z = x + iy = U and v are expressed in terms of r and θ. Differentiate it partially w.r.t. r and θ, we get By equating real and imaginary parts, we get 
Key takeaways
 A function is said to be analytic at a point if f is differentiable not only at but every point of some neighbourhood at .
 A point at which the function is not differentiable is called singular point.
 A function which is analytic everywhere is called an entire function.
 If a function is analytic in a domain D, then u and v will satisfy CauchyRiemann conditions.
 CR conditions are necessary but not sufficient for analytic function.
 CR conditions are sufficient if the partial derivative are continuous
In Cartesian form
Theorem; The necessary condition for a function to be analytic at all the points in a region R are
(ii) Provided, 
Proof:
Let be an analytic function in region R. Along real axis Then f’(z), becomes ………… (1)
Along imaginary axis From equation (1) and (2) Equating real and imaginary parts Therefore and These are called Cauchy Riemann Equations. 
CR equation in polar from
CR equations in polar form are
Proof:
As we know that x = r cos and u is the function of x and y z = x + iy = r ( cos Differentiate (1) partially with respect to r, we get Now differentiate (1) with respect to , we get Substitute the value of , we get Equating real and imaginary parts, we get Proved 
Key takeaways
(ii) 2. CR equations in polar form are 
Necessary condition for function f(z) to be analytic
Theorem; The necessary condition for a function to be analytic at all the points in a region R are
(ii) Provided, 
Proof:
Let be an analytic function in region R. Along real axis Then f’(z), becomes ………… (1)
Along imaginary axis From equation (1) and (2) Equating real and imaginary parts Therefore and These are called Cauchy Riemann Equations. 
Sufficient condition for function f(z) to be analytic
Theorem
The sufficient conditions of a function f(z) = u + iv to be analytic at all the points in the region R are
1. 2. are continuous functions of x and y in region R. 
Proof:
Suppose f(z) be a singlevalued function which has At each point in the R region, then the CauchyReimann equations are satisfied. By Taylor’s theorem Ignore the terms of higher power. We know that from CR equations Replace by We get Proved 
Conformal mapping
If the sense of the relation as well as magnitude of the angle is preserved the transformation is said to be conformal.
NoteAn analytic function f (z) is conformal everywhere except at its critical points where f (z) = 0.
Example1: Find the conformal transformation of .
Answer. Let 
Theorem: If W=f(z) represents a conformal transformation of a domain D in the zplane into a domain D of the W plane then f(z) is an analytic function of z in D.
Proof:
We have u+iv=u(x,y)+iv(x,y) So that u=u(x, y) and v=v(x,y) Let ds and denote elementary arc length in the zplane and wplane respectively Then Now Hence Or Where Now is independent of direction if Where h depends on x and y only and is not zero. Thus the conditions for an isogonal transformation And The equation are satisfied if we get Then substituting these values in 2 we get Taking i.e. Also Hence Similarly i.e. The equation (4) are the wellknown Cauchy Reimann Conformal mapping 
Example: Show that the mapping is conformal in the whole of the z plane.
Sol.
Let z=x+iy Then 
Consider the mapping of the straightline x=a in z plane the w plane which gives which is a circle in the w plane in the anticlockwise direction similarly the straightline y=b is mapped into which is a radius vector in the w plane.
The angle between the line x=a and y=b in the z plane is a right angle. The corresponding angle in the w plane between the circle e = constant and the radius vector is also a right angle which establishes that the mapping is conformal.
Example: Show that the curve u = constant and v = constant cut orthogonally at all intersections but the transformation w = u + iv is not conformal. Where
Sol.
Let …………. (1) Differentiate (1), we get …………… (2) Now …………….. (3) Differentiate (3), we get ………. (4) As we know that for the condition for orthogonallity, from (2) and (4) So that these two curves cut orthogonally. Here, And Here the CR equation is not satisfied so that the function u + iv is not analytic. Hence the transformation is not conformal. 
Key takeaways
 If the sense of the relation as well as magnitude of the angle is preserved the transformation is said to be conformal.
 An analytic function f (z) is conformal everywhere except at its critical points where f (z) = 0
Transformation:
Sol. Now equating real and imaginary parts, we get Case1: if and then shows equilateral hyperbolas with the lines y = ± x and the coordinate axes x = 0, y = 0 as asymptotes respectively) which are orthogonal trajectories of each other.
Case2: if and then eliminating x and y, we get So that Which is a parabola with focus at origin, v = 0 As axis and open to the left. Similarly So that These parabolas are orthogonal to each other. w = z2 is conformal everywhere except at z = 0 where w = 2z = 0.
Transformation: Now equating real and imaginary parts Again Hence
Transformation: At z = So the transformation is not conformal at z =
and Now

Bilinear transformation is a correction of backwards difference method.
The bilinear transformation (also known as Tuatn’s method transformation) is defined as substitution:
Example 1:
Find the bilinear transformation which aps points z=2,1,0 ontpo the points w=1,0,i
Sol.
Let Thus we have = 
Example 2:
How that the bilinear transformation
w= transforms 
in the zplane to 4u+3=0 in wplane.
Sol.
Consider the circle in zplane = 0 Thus, centre of the circle is (h,k)c(2,0) and radius r=2. Thus in zplane it is given as =2....(1) Consider w= W(z4) = 2z+3 Wz4w=2z+3 Wz2z=4w+3 Z(w2) = (4w+3) z = z2 =  2 
References
 E. Kreyszig, “Advanced Engineering Mathematics”, John Wiley & Sons, 2006.
 P. G. Hoel, S. C. Port And C. J. Stone, “Introduction To Probability Theory”, Universal Book Stall, 2003.
 S. Ross, “A First Course in Probability”, Pearson Education India, 2002.
 W. Feller, “An Introduction To Probability Theory and Its Applications”, Vol. 1, Wiley, 1968.
 N.P. Bali and M. Goyal, “A Text Book of Engineering Mathematics”, Laxmi Publications, 2010.
 B.S. Grewal, “Higher Engineering Mathematics”, Khanna Publishers, 2000.
 T. Veerarajan, “Engineering Mathematics”, Tata McgrawHill, New Delhi, 2010
 Higher engineering mathematics, HK Dass