I had a complex analysis exam yesterday which had some multiple choice questions in it, I want to double check my answers to see if I got some right. I have typed up the questions and my ansers for them.
- Which of the following is false?
a) $e^z = \cos{z} + i \sin{z}$ with $z∈\mathbb{C}$
b) $ z^w = e^{wlogz}$ with $z,w∈\mathbb{C}$
c) $log(−1) = iπ + 2πik$ with $k∈\mathbb{Z}$
d) $ e^{z−iπ} = i^2e^z$ with $z∈\mathbb{C}$.
e) None of the above. ( I chose this one)
- Which of the following functions is equal to the power series $\sum^{\infty}_{n=0}{\frac{(-1)^n}{n!}z^n}$
b) $z→e^{-z}$ with $z∈\mathbb{C}$ (I chose this)
- Which one of the following functions is not holomorphic?
d) $f: \mathbb{C} \to \mathbb{C}$ with $z \to \log{z}$. (I chose this)
- Let $ f: \mathbb{C} \to \mathbb{C}$ be holomorphic. Which one of the following statements is false?
d) $f$ must be unbounded. (I chose this)
- What is the value of the contour integral $\frac{1}{2\pi i} \oint_y \frac{f(C)}{(C-z)^2}dC$, where $f: \mathbb{C} \to \mathbb{C}$ is holomorphic and $y$ a closed circle with centre at $z\in \mathbb{C}$, which is traced anticlockwise?
b) $f'(z)$. (I chose this)
- Which one of the following functions is a harmonic conjugate of $u(x,y)= x^2 -y^2$?
c) $v(x,y)=2xy$ (I chose this)
- Consider the path $γ : [0,1] → \mathbb{C}$ with $t → −i + 2e^{2πit}$. On which one of the following open sets is $γ$ null-homotopic, i.e. homotopic to a trivial path?
a) $Ω = \{ z∈\mathbb{C} : 1/2 <|z|< 4 \}$. (I chose this)
- Let $c∈ \mathbb{C}$ be an isolated singularity of the holomorphic function $f : \mathbb{C} \backslash \{c\} → \mathbb{C}$ and $γ$ a small circle centered at $c$ traced anticlockwise. Which one of the following is false?
a) If $|f(z)| \to \infty$ as $z \to c$ then $c$ is a pole. (I chose this)
- What is the residue of the function $f(z) = e^{−z}/z^n$ with $n > 0$ an integer?
a) $n$
b) $(-1)^nn$
c) $(-1)^n n!$
d)$(-1)^n / n!$
e) None of the above. (I chose this)
- Consider the real integral $I= \int^{\infty}_{- \infty} \frac{1}{1+x^2} dx$. The latter can be computed using Cauchy's Residue Theorem. Which one of the following statements is false?
a) Let $γ(R)$ be a semicircle of radius $R > 1$ traced anticlockwise, centred at the origin and closed in the lower half of the complex plane. Then $I = lim_{R \to \infty} \oint_{γ(R)} \frac{1}{1+z^2}dz$.
b) Let $γ(R)$ be a semicircle of radius $R > 1$ traced clockwise, centred at the origin and closed in the lower half of the complex plane. Then $I = lim_{R \to \infty} \oint_{γ(R)} \frac{1}{1+z^2}dz$.
c) $I= 2 \pi i \text{Res}(\frac{1}{1+z^2}, z=i)$
d) $I=-2 \pi i \text{Res}(\frac{1}{1+z^2}, z=-i)$
e) None of the above. (I chose this)
1) I guess the false one is the first one as for $z=\pi/2$ it gives you $e^{\pi/2} = i$ that is not true. Maybe there's a typo here and you meant $e^{iz}$ ? In that case, the question is ambiguous but none of the statements is wrong if you're not too rigorous.
2) you're right
3) you're right
4) you're wrong without any further assumptions : a constant function over $\mathbb{C}$ is holomorphic
5) I think it's the opposite (with a $-$) but I may be wrong, I didn't do the full computation
6) Seems right
7) I think you're wrong : as the circle $\gamma$ is "inside" the annulus $\Omega$, there's no way it is homotopic to a point here.
8) You're right
9) I would say $\dfrac{(-1)^{n-1}}{(n-1)!}$ so yeah, none of the above
10) edit$^2$ : I think the last statement is false : the $-$ should not be there, because for every simple anticlockwise curve $\gamma$, $2i\pi * \mathrm{Res} = \int_\gamma f(z)\mathrm{d}z$, and here with $\gamma$ the second curve (the one under the real axis) there is no reason for this $-$ sign to appear
edit^3 : in fact I think as the integral is taken from the right to the left by choosing this path $\gamma$, you have to change the sense, an then add a $-$. So the last statement looks true, and none of the above statement are false. Sorry!