(a) Give the definition of $e^z$ for a complex number $z = x+iy$ (2 marks)
(b) Use the Cauchy-Riemann equations to prove that $f\colon \mathbb C \to \mathbb C$, $f(z) = e^{2z+i}$ is differentiable at every point of $\mathbb C$, and that $f'(z) = 2f(z)$. (6 marks)
(c) Explain why the function $f(z) = e^{2z+i}$, $z \in \mathbb C$, is analytic at all points of $\mathbb C$. (2 marks)
(d) Determine the value of the integral $\int_\gamma e^{2z+i}\, dz,$ where $\gamma$ is the triangle in $\mathbb C$ with vertices in the 3rd roots of $1+i$, oriented clockwise. (5 marks)
In part (d) of this question...As the function $e^{2z+i}$ is analytic everywhere in the complex plane, particularly on and inside the curve $\gamma$ by Cauchy's Theorem $\int e^{2z+i}dz$ = 0. Is that correct...I dont have to bother with any of the triangle related stuff?
And for (c), I have proved in (b) that this function is differentiable everywhere in C so I can't see why I am being asked why it is analytic at all point in C. Am I basically just supposed to repeat what I found in (b)? That f(z) is analytic at all points of C as because it is differentiable at all points of C, as it is a composition of analytic functions?