What are the Cauchy-Riemann equations used to determine?

The partial derivatives of complex functions.

Fill in the blank: A complex function is said to be ______ if it is complex differentiable at every point in a region.

Analytic

True or False: The Cauchy-Riemann equations must be satisfied for a function to be holomorphic.

True

Riddle: I connect real and imaginary, ensuring smooth transitions in the complex plane. What am I?

Cauchy-Riemann equations

In polar coordinates, the Cauchy-Riemann equations facilitate the analysis of complex functions in a more ______ manner.

Versatile

Which of the following expressions relates to the Cauchy Integral Theorem? A) ∫C f(z) dz = 0 B) f(z) = u + iv C) ∂u / ∂x = ∂v / ∂y D) All of the above

A) ∫C f(z) dz = 0

Fill in the blank: The imaginary part of the analytic function f(z) = x² - y² + iψ(x, y) at z = (1 + i) is ______.

2

True or False: A region is simply connected if it contains holes.

False. A simply connected region has no holes.

The value of k that makes the complex-valued function analytic, where z = x + iy, is ______.

1.999 - 2.001

Riddle: I extend a function beyond singularities, allowing for negative powers. What am I?

Laurent series