Question

Which of the following is correct?

• A. The Cauchy-Riemann equations are given by $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \, \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.$
• B. The function $f(z) = z$ is analytic everywhere.
• C. If $f(z)$ is analytic inside and on a simple closed curve $C$ except at a finite number of points $a, b, c, \dots$ inside $C$, at which the residues are $\text{Res}(f, a), \text{Res}(f, b), \text{Res}(f, c), \dots$, then $$\oint_C f(z) \, dz = 2\pi i (\text{Res}(f, a) + \text{Res}(f, b) + \dots).$$
• D. If $f(z)$ is analytic inside and on a simple closed curve $C$ and $a$ is any point inside $C$, then $$f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - a} \, dz.$$

Hint:

The Cauchy integral formula is fundamental in complex analysis and provides insights into function behavior within analytic regions.

Answer 1

The Correct Option is D

The statement in option 4 represents the Cauchy integral formula, which relates the value of a function at a point to a contour integral around that point.