Question

Problem 2 Suppose γ:R→C is defined by γ(t)=t+it Sketch - the image of R under γ(t). - the image of R under eγ(t). - the image of R under log(eγ(t)). Explain (briefly) your sketches. Problem 3 Suppose a function f:C→C is called antiholomorphic at z if the limit limh→0​hˉf(z+h​)−f(zˉ)​ exists. What is the analog of the Cauchy-Riemann equations for antiholomorphic functions?

Answer 1

In problem 2, we are given the function γ(t) = t + it, and we need to sketch the image of the real line under this function, the image of the real line under eγ(t), and the image of the real line under log(eγ(t)).

Step-by-step explanation:

In problem 2, we are given the function γ(t) = t + it, and we need to sketch the image of the real line under this function, the image of the real line under eγ(t), and the image of the real line under log(eγ(t)).

For the function γ(t) = t + it, the image of the real line will be a line in the complex plane passing through the origin.

For the function eγ(t), the image of the real line will be the unit circle in the complex plane.

For the function log(eγ(t)), the image of the real line will be a line in the complex plane passing through the point (1, 0).