Review of Complex Numbers A complex number z = (x,y) is an ordered pair of real numbers; x is called the real part and y is called the imaginary part, or x = Re z and y = Im z. Two complex numbers are equal if and only their real and imaginary parts are equal. (0,1) is called the imaginary unit, denoted by i. Addition and multiplication are defined as follows: –z 1 + z 2 = (x 1,y 1 ) + (x 2,y 2 ) = (x 1 + x 2, y 1 + y 2 ) –z 1 z 2 = (x 1 x 2 – y 1 y 2, x 1 y 2 + x 2 y 1 ) With the above rules, we can essentially identify the real numbers with complex numbers of the form (x,0) and for any real y, (0,y) = iy. We also see that i 2 = – 1. Therefore we write z = x + iy, and do multiplication by ordinary real arithmetic but replacing i 2 by – 1 wherever it occurs.

Geometry of Complex Numbers We represent complex numbers by points in a plane (analogous to ordered pairs of real numbers), i.e. the complex number z = x + iy is represented by the point (x,y) in the complex plane. When we do this, addition of complex numbers is similar to addition of vectors in the plane. The distance of the complex number z from the origin is then given by  x 2 + y 2, written |z|, and called the absolute value or modulus of z. If z = x + iy is a complex number, then its complex conjugate is the complex number x  iy, written  z. We see that |z| 2 = z  z.

Polar Form and De Moivre’s Formula We can represent complex numbers in polar form by putting x = r cos , y = r sin  so that z = x + iy = r(cos  + i sin  ), where r =  x 2 + y 2 = |z|, and  = arg z = arc tan (y/x), called the argument of z, which is defined as the directed angle with the positive real axis. We see that z n = r n (cos  + isin  ) n = r n (cos n  + isin n  ) for n = 0,1,2,3,…. A particular case of this is: De Moivre’s Formula: (cos  + isin  ) n = (cos n  + isin n  ) This can be extended to negative integral powers also.

COMPLEX FUNCTIONS A complex function w = f(z) is a rule that assigns to every z in a set S a complex number w called the value of f at z. The set S is called the domain of f. We usually take the domain of f to be a domain, that is an open connected set. The set of output values is called the range of the function f. Since w is complex, we may write w = u + iv, where u and v are the real and imaginary parts respectively. Now w depends on z = x + iy. Hence, u and v become real functions of x and y, and we can write w = f(z) = u(x,y) + iv(x,y). Thus a complex function f(z) is equivalent to a pair of real functions of two variables, referred to as Re f and Im f (the real and imaginary parts of f).

Limit of a Function This is defined analogously to the case for real functions. Informally: Let f(z) be defined on an open region about z 0, except possibly at z 0 itself. If f(z) gets as close as desired to L for all z close to z 0, we say that f approaches the limit L as z approaches z 0. Formal Definition: If given any  > 0, there exists a  > 0 such that |f(z) – L| <  for all z in the disk |z – z 0 | <  except at z 0 itself, we say that f approaches the limit L as z approaches z 0 : lim f(z) = L z  z 0

Behaviour of Limits Proposition 1: Suppose limits of functions f and g at z 0 are L and M. Then: a)lim (f  g) = L  M b)lim (cf) = cL, where c is constant c)lim (f.g) = L.M d)lim(f/g) = L/M, provided M  0

Continuity of Functions We say that f(z) is continuous at z 0 if lim f(z) = f(z 0 ) z  z 0 Remark: as in the case of functions of two variables, we allow z to approach z 0 from any direction in the complex plane. f(z) is said to be continuous in a domain if it is continuous at every point in this domain.

Similarity of Concepts of Limits and Continuity Proposition 2: Suppose functions f and g are continuous at z 0. Then the following functions are also continuous at z 0 : a) f  g b)cf, where c is constant c)f.g d)f/g, provided g(z 0 )  0 e)the composed function g  f, defined by (g  f)(z) = g(f(z)), is also continuous at z 0, provided g is defined in a neighbourhood of f(z 0 ) and continuous at f(z 0 )

The Derivative The derivative of the function f at a point z 0 is written f’(z 0 ), and is defined by : f’(z 0 ) = lim f(z 0 +  z) – f(z 0 ) = lim f(z) – f(z 0 )  z  0  z z  z 0 z – z 0 provided the limit exists Remark: as in the case of functions of two variables, we allow z to approach z 0 along any path, and the limit must be same along all the paths. This imposes some very strong consequences in the case of complex functions. Remark: If f is differentiable (i.e. has a derivative) at a point, then it is continuous at that point (left as an exercise).

Properties of Derivatives Proposition 3: The differentiation rules for complex functions are the same as for real functions: : Sum/difference rule: (f  g)’ = f’  g’ Product rule: (fg)’ = fg + fg’ Quotient rule: (f/g)’ = (gf’ – fg’)/g 2 Chain rule: (f  g)’(z) = f’(g(z))g’(z) Power rule: (z n )’ = nz n – 1

Analytic Functions A function f(z) is said to be analytic in a domain D if f(z) is defined and differentiable at every point of D. f(z) is said to be analytic at a point z = z 0 if f(z) is differentiable in some neighbourhood of z 0. Another term for analytic in D is holomorphic in D. Functions which are analytic/holomorphic in the entire complex plane are said to be entire functions.

Cauchy-Riemann Equations Theorem 1(a) (Necessity) : Let w = f(z) = u(x,y) + iv(x,y) be a complex function which is defined and continuous in some neighbourhood of z = x + iy and differentiable at z itself. Then at that point, the first order partial derivatives of u and v exist, and satisfy the Cauchy Riemann equations: u x = v y and u y = – v x ……. (1) Hence, if f(z) is analytic in a domain D, the partial derivatives exist and satisfy (1) at all points of D.

Differentiation Rules Proposition 4: The following rules are useful for finding out the derivative of a complex function f(z):  f’(z) = u x + iv x  f’(z) = v y – iu y  f’(z) = u x – iu y  f’(z) = v y + iv x However, first check by using Cauchy-Riemann equations whether the given function is analytic.