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.

2. 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.

3. 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.

4. Cauchy-Riemann Equations - 2 Theorem 1(b) (Sufficiency): If two real-valued functions u(x,y) and v(x,y) have continuous first partial derivatives in some domain D containing a point z 0, and they satisfy the Cauchy-Riemann equations at z 0, then w = f(z) = u(x,y) + iv(x,y) has a derivative f’(z 0 ) at z 0. Remark: Proof is difficult and will be omitted. As a corollary to the above, the function f(z) is analytic in a domain D if u(x,y) and v(x,y) have continuous first partial derivatives in D and satisfy the Cauchy-Riemann conditions throughout D. Remark: The Cauchy-Riemann equations can be used to find out whether any given complex function is analytic or not. However, care has to be exercised since the sufficiency conditions are stronger.

5. Major Property of Analytic Functions Proposition 5: If f(z) is analytic in a domain D, then it has derivatives of all orders in D, which are then also analytic functions in D. Remark: Proof of the above requires Cauchy’s integral formula. We will look into the proof later. The following corollary is also of importance: Corollary 5.1: if f(z) = u(x,y) + iv(x,y) is analytic in a domain D, then u and v both have continuous second partial derivatives in D and both satisfy Laplace’s Equation:  2 u = 0 and  2 v = 0 Remark: Proof of colollary is left as an exercise. A real-valued function g(x,y) of two variables which satisfies Laplace’s Equation is said to be a harmonic function. In other words, if f(z) = u(x,y) + iv(x,y) is analytic, both its real and imaginary parts (i.e. u and v) are harmonic functions. In this case, they also have to satisfy C-R conditions, and so are known as conjugate harmonic functions.

6. Counter-Example of Interest Define f(z) = [x 3 (1 + i) – y 3 (1 – i)]/(x 2 + y 2 ), for z  0 = 0, at z = 0 Then, at z = 0, u x = 1, v y = 1, and u y = – 1, v x = 1 So Cauchy-Riemann Conditions are satisfied But f(z) is not differentiable at the origin This is because Theorem 1(b) is not satisfied at the origin