List of top Mathematics Questions on Complex Analysis asked in CPET

Which one of the options given below is false?

For an entire function \(f\), which one of the following statements is false?

If \(\displaystyle\sum_{n=-\infty}^{\infty}a_nz^n\) is the Laurent series expansion of the function \(f(z)=\dfrac{1}{2z^2-13z+15}\) in the annulus \(\{z\in\mathbb{C}:3/2<|z|<5\}\), then \(a_1/a_2=\) ____.

Which of the following statements is/are true?
(I) The function \(f(z)=\sin\left(\frac{1}{\cos(1/z)}\right)\) has an isolated singularity at \(z=0\).
(II) The value of the integral \(\oint_{|z|=1}\frac{dz}{z\sin(z)}\) is zero.

Consider the following statements:
(I) Residue of \(f(z)=e^z\) at \(z=\infty\) is \(-1\).
(II) If a meromorphic function \(f\) has 5 simple zeros and 2 simple poles inside \(|z|=1\), then \(\oint_{|z|=1}\frac{f'(z)}{f(z)}dz = 6i\pi\).
Pick out the correct option.

Which of the statements given below is/are false?
(I) \(f(z)=x^2+y^2+2ixy\) (\(z=x+iy\)) is differentiable only at the points that lie on the x-axis.
(II) \(f(z)=|z-1|^2\) is differentiable at \(z=1\), but not analytic at \(z=1\).
(III) There exists an analytic function in \(\mathbb{C}\) whose imaginary part is \(x^2\).

Which one of the following options is true?
(A) If \(\operatorname{Log}(z)\) denotes the principal value of the logarithm, then \(\operatorname{Log}(zw)=\operatorname{Log}(z)+\operatorname{Log}(w)\) for all \(z,w\in\mathbb{C}\setminus\{0\}\).
(B) The solution set of the equation \(e^{iz}=-1\) is \(\{(2k+1)\pi:k\in\mathbb{Z}\}\).
(C) \(\cos(z)\) is bounded in the whole complex plane.
(D) A Mobius transformation with three fixed points is a constant.

The radius of convergence of the power series \(\sum_{n=1}^\infty\left(\frac{2^n}{n}+\frac{3^n}{n^2}\right)z^n\) is ____.

If \(C\) is a simple closed curve not passing through the points \(-1\), \(0\) and \(1\) in the complex plane, then the set of all possible value(s) of the integral \(\int_C \dfrac{dz}{z(1-z^2)}\) is ____.

Consider the following statements:
(I) The function \(f(z)=\dfrac{x^2-y^2+2ixy}{x^2+y^2}\) (\(z=x+iy\)) has a limit as \(z\to0\).
(II) The function \(f(z)=\begin{cases}\operatorname{Re}(z)/|z|, & z\ne0\\ 1, & z=0\end{cases}\) is continuous at \(z=0\).
Choose the correct answer: