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(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
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2 Answers 2

According to your own definition of "analytic function", it looks like (b) and (c) are completely identical...

About (d): many times Cauchy's Theorem is proved first for very special paths (circles, triangles, rectangles, etc.). In its most general form, this theorem allows to deduce at once that the integral equals zero.

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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.

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