# Complex integral revision, this is just Cauchy's Theorem right?

(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?

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Can you please edit the questions into the question, rather than having it in an image? – Aryabhata Apr 17 '12 at 21:14
Why, what's wrong with an image? – Jim_CS Apr 17 '12 at 21:16
It is not full text searchable: see: meta.math.stackexchange.com/questions/1805/…. We have math support on this site (and you have used it before). btw, what is the definition of Analytic function that you are using in the class? – Aryabhata Apr 17 '12 at 21:18
"A function f of the complex variable z is analytic at a point $z_0$ if it has a derivative at each point in some neighbourhood of $z_0$" – Jim_CS Apr 17 '12 at 21:55
Seems like you have solved it. Perhaps you need to ask the professor involved about part c). – Aryabhata Apr 17 '12 at 22:24

For part (d) you don't need Cauchy's theorem. It suffices to note that the function $f(z):=e^{2z+i}$ is the derivative of the function $F(z):={1\over2}e^{2z+i}$. The integral of a derivative $F'$ along any curve from $p$ to $q$ is equal to $F(q)-F(p)$; therefore the integral of $F'$ along any closed curve is zero.