Complex Analysis Complex Numbers:

A complex number z is an ordered pair (x,y) of real numbers x and y. z = (x,y) = x + iy
The Real Part of z ie.Re(z) = x and the Imaginery part of z ie. Im(z) = y.
Moreover,i2 = -1 which is an imaginery unit. a. The two imaginery numbers x + iy and a + ib are equal iff x = a and y =b, b. For z = x + iy, if x = 0,then z = iy (A pure imaginery number) and if y = 0 then z = x ( Pure real number).
If z1 = x1 + iy1 and z2 = x2 + iy2, then the Addition, Multiplication, Subtraction and Division of two complex numbers respectively is defined as follows: z1 + z2 = (x1 + x2) + i(y1 + y2) z1 z2 = (x1 x2 - y1 y2) + i(x1 y2 + x2 y1) z1 − z2 = (x1 - x2) + i(y1 - y2) z = x/y = x + iy,where x = , y = ,z2 ≠ 0. Complex Conjugate Number
The complex conjugate of the number z = x +iy is = x-iy Re(z) = x = (z + ) and Im(z) = (z - ) When z is real, z = x then z =

Polar Form of Complex Numbers
Let (x,y) be the Cartesian coordinates and (r,Ө) be the polar coordinates,then x = r cos Ө , y = r sin Ө
Therefore, z = x+iy = r (cos Ө+ isin Ө) r = which is the absolute value or the modulus of z. Ө = arg z = tan which is the argument of z. Important Properties Generalized Triangle Inequality : Let Then,

De Moivre’s formula : Nth Root of z : Limit, Continuity and Derivatives of Function of Complex variable:

Limit : Let the function of a complex variable : w = f(z) = f(z+iy) = u(x,y)+iv(x,y). A function f(z) has a limit l at if exists.
Continuity : A function f(z) has a continuity at z0 , if f(z0) is defined and Derivative : A function f(z) is differentiable at z0 , if exists.
Moreover, f(z) has a derivative at z0. If the function is differentiable at z0, then it is also at z0.

Analytic Function :
A function f(z) is analytic in a domain D if it is differentiable everywhere in and on the domain D. A function f(z) is analytic at z0 if it is differentiable at each point in the neighbourhood of z0 including itself. A point where the function f(z) is not analytic is known as a singular point. Cauchy Riemann Equation :
If f(z) = u(x,y) + iv(x,y) , then Cauchy Reimann Equation is given by : ux = vy and uy = - vx. Laplace Equation for : = 0 Some Important Theorems :

If f(z) = u(x,y) + iv(x,y) is analytic in domain D , then it satisfies Cauchy Riemann equation and its partial derivatives exist i.e. ux = vy and uy = - vx.. If f(z) = u(x,y) + iv(x,y) is analytic in domain D, then u and v satisfies Laplace equation i.e. = 0

= 0
The function becomes the harmonic function when

Important Definitions :

Circle,disk and half plane: Unit circle : Circle of radius ‘ ’and centre ‘a’: Open circular disk : Closed circular disk : Open annulus : Closed annulus : Upper half plane : Lower half plane : Right half plane : Left half plane : Open set, closed set and boundary points : Neighbourhood of a point ‘a’ is , an open circular disk of radius A set S is open if has a neighbourhood , that entirely consists of points belongs to S. A set is connected if, and the points of line segment that joins x and y also belongs to S. A complement of a set S is the set of all points of a complex plane that does not belongs to S. The complement of a open set is closed set. A Boundary point of a set S is a point, whose neighbor contains the points that belongs to S and the points that does not belong to S. Domain is an open connected set. Curve : Curve C is defined as C: where x and y are continuous function and the sense of increase in t is called positive sense on C. C is simple if it does not cross itself. C is closed if the starting point is itself the end point. C is differentiable if x(t) and y(t) are differentiable for . C is contour, if C is formed by joining finitely many simple smooth curves end to end.

Cauchy Integral Theorem :

Cauchy Integral Theorem : If f(z) is analytic in a simple connected domain D, then for every simple closed path C in D, Independence of path : If f(z) is analytic in a simple connected domain D, then the integral of f(z) in independent of path in D. Deformation of contour : Let C1 and C2 be two simple closed positively oriented contours such that C1 lies inside C2 in domain D.
If F is analytic in a domain D that contains both C1 and C2 and the region between them, then Extended Cauchy’s Theorem : Let C1,C2,………,Cn be simple closed positively oriented contours with each Ck lies interior to C for k = 1,2,……,n and the set interior to Ck has no point in common with set interior to Ci if k ≠ i. Let f be analytic on domain D, that contains all the contours and the region between C and C1 +C2+………+Cn, then

Cauchy’s Integral Formula :

Let f(z) be analytic in a simple connected domain D and let C be a simple closed positively oriented contour that lies in D. Then for any point z0 Є D and z0 lies interior to C,

Cauchy’s Integral Formula for derivatives :

Let f(z) be analytic in the simple connected domain D, and let C be a simple closed positively oriented contour that lies in D. Then for any point point z0 Є D and z0 lies interior to C,

Morera’s Theorem :

If f(z) is continuous in a simple connected domain D and if for every closed path in D,then f(z) in analytic in D. It is the converse of Cauchy’s Theorem.

Liouvelle’s Theorem :

If an entire function f(z) is bounded in absolute value for ,then f(z) must be a constant.

Cauchy’s Inequality :

Let f be analytic in the simple connected domain D, that contains the circle C: . If holds for all z on C, then

Exercise : Let z2 = 3 + 5i and z1 = 4 + 3i then find ? Let z1 = 2 + 4i and z2 = 6 + 7i then prove that Re(z1) + Re(z2) = Re(z1+z2) Give the polar representation of 6 + 2i? Let f(z) = z2 + 3z, then find the value of Re(z) and Im(z)? What is the modulus of and argument of 1 - cosα + isinα? Show that f(z) = log z satisfies Cauchy – Riemann Equation but is not analytic at z = 0. Prove that where C is a circle Evaluate where C is the circle , , Evaluate where C is the circle . Prove that the functions u = is harmonic and find its harmonic conjugates. Prove that u = cos x cosh y is harmonic function and find its harmonic conjugates. Prove that ex(cos y + isin y) is holomorphic and find its derivative. Prove that f(z) = sin x (cosh y) + icos x (sinh y) is continuous as well as analytic everywhere. Show that the function f(z) = xy + iy is everywhere continuous but is not analytic. If f(z) is an analytic function of z, prove that State and prove Liouvelle’s Theorem.