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.