Question

1. (a)Express Cauchy-Riemann's equations as the condition for differentiability of a
complex function. Show that f(2)=z' is complex differentiable.

Answer 1

Answer:

Analytic Functions

Analytic FunctionsIf f (z) is differentiable and single-valued in a region of the complex plane, it is said to be an analytic function in that region.1 Multivalued functions can also be analytic under certain restrictions that make them single-valued in specific regions; this case, which is of great importance, is taken up in detail in Section 11.6. If f (z) is analytic everywhere in the (finite) complex plane, we call it an entire function. Our theory of complex variables here is one of analytic functions of a complex variable, which points up the crucial importance of the Cauchy-Riemann conditions. The concept of analyticity carried on in advanced theories of modern physics plays a crucial role in the dispersion theory (of elementary particles). If f′(z) does not exist at z = z0, then z0 is labelled a singular point; singular points and their implications will be discussed shortly.

belled a singular point; singular points and their implications will be discussed shortly.To illustrate the Cauchy-Riemann conditions, consider two very simple examples.

belled a singular point; singular points and their implications will be discussed shortly.To illustrate the Cauchy-Riemann conditions, consider two very simple examples.Example 11.2.1

belled a singular point; singular points and their implications will be discussed shortly.To illustrate the Cauchy-Riemann conditions, consider two very simple examples.Example 11.2.1z2 Is Analytic

belled a singular point; singular points and their implications will be discussed shortly.To illustrate the Cauchy-Riemann conditions, consider two very simple examples.Example 11.2.1z2 Is AnalyticLet f (z) = z2. Multiplying out (x − iy)(x − iy) = x2 − y2 + 2ixy, we identify the real part of z2 as u(x, y) = x2 − y2 and its imaginary part as v(x, y) = 2xy. Following Eq. (11.9),

belled a singular point; singular points and their implications will be discussed shortly.To illustrate the Cauchy-Riemann conditions, consider two very simple examples.Example 11.2.1z2 Is AnalyticLet f (z) = z2. Multiplying out (x − iy)(x − iy) = x2 − y2 + 2ixy, we identify the real part of z2 as u(x, y) = x2 − y2 and its imaginary part as v(x, y) = 2xy. Following Eq. (11.9),We see that f (z) = z2 satisfies the Cauchy-Riemann conditions throughout the complex plane. Since the partial derivatives are clearly continuous, we conclude that f (z) = z2 is analytic, and is an entire function.