Complex Analysis Self-Test

Complex Analysis Quiz

Test your knowledge of complex analysis concepts

Q1

The polar form of a complex number $z = x + iy$ is:

$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ where $r = |z|$ and $\theta = \arg(z)$ $z = r(\sin\theta + i\cos\theta) = re^{i\theta}$ where $r = |z|$ and $\theta = \arg(z)$ $z = r(\cos\theta - i\sin\theta) = re^{-i\theta}$ where $r = |z|$ and $\theta = \arg(z)$ $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ where $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}(y/x)$

Q2

The Cauchy-Riemann equations for a complex function $f(z) = u(x,y) + iv(x,y)$ to be analytic are:

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ $\frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$ $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial x}$ and $\frac{\partial u}{\partial y} = \frac{\partial v}{\partial y}$ $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$

Q3

A function $f(z)$ is analytic at a point $z_0$ if:

It is differentiable in some neighborhood of $z_0$ It is differentiable only at $z_0$ It is continuous at $z_0$ It satisfies the Cauchy-Riemann equations at $z_0$

Q4

An entire function is:

A function that is analytic everywhere in the complex plane A function that is analytic in some domain A function that is differentiable at a point A function that is continuous everywhere

Q5

The order of an entire function $f(z)$ is defined as:

$\rho = \inf\{\lambda > 0 : |f(z)| \leq e^{|z|^\lambda} \text{ for } |z| \text{ sufficiently large}\}$ $\rho = \sup\{\lambda > 0 : |f(z)| \leq e^{|z|^\lambda} \text{ for } |z| \text{ sufficiently large}\}$ $\rho = \lim_{r \to \infty} \frac{\ln \ln M(r)}{\ln r}$ where $M(r) = \max_{|z|=r} |f(z)|$ Both (a) and (c)

Q6

A necessary and sufficient condition for a function $f(z) = u(x,y) + iv(x,y)$ to be analytic in a domain $D$ is:

The first-order partial derivatives of $u$ and $v$ exist, are continuous, and satisfy the Cauchy-Riemann equations in $D$ $u$ and $v$ are harmonic functions in $D$ $u$ and $v$ satisfy the Cauchy-Riemann equations in $D$ $f$ is continuous in $D$ and satisfies the Cauchy-Riemann equations in $D$

Q7

A real-valued function $u(x,y)$ is harmonic in a domain $D$ if:

It has continuous second-order partial derivatives and satisfies Laplace's equation $\nabla^2 u = 0$ in $D$ It is the real part of an analytic function in $D$ It satisfies the Cauchy-Riemann equations in $D$ Both (a) and (b)

Q8

If $u(x,y)$ is harmonic in a simply connected domain $D$, then its harmonic conjugate $v(x,y)$ is:

A function such that $f(z) = u + iv$ is analytic in $D$ A function such that $f(z) = v + iu$ is analytic in $D$ A function such that $u$ and $v$ satisfy the Cauchy-Riemann equations A function such that $v$ is also harmonic in $D$

Q9

The contour integral $\int_C f(z) dz$ along a contour $C$ parametrized by $z(t) = x(t) + iy(t)$, $a \leq t \leq b$, is defined as:

$\int_a^b f(z(t)) z'(t) dt$ $\int_a^b f(z(t)) |z'(t)| dt$ $\int_a^b f(z(t)) dt$ $\int_C f(z) |dz|$

Q10

Cauchy's integral theorem states that if $f$ is analytic in a simply connected domain $D$ and $C$ is any simple closed contour in $D$, then:

$\int_C f(z) dz = 0$ $\int_C f(z) dz = 2\pi i f(z_0)$ where $z_0$ is any point inside $C$ $\int_C f(z) dz = 2\pi i \sum \text{Res}(f, z_k)$ where $z_k$ are the singularities inside $C$ $\int_C f(z) dz = f'(z_0)$ where $z_0$ is any point inside $C$

Q11

Cauchy's integral formula states that if $f$ is analytic inside and on a simple closed contour $C$ and $z_0$ is inside $C$, then:

$f(z_0) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - z_0} dz$ $f'(z_0) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - z_0} dz$ $f(z_0) = \frac{1}{2\pi} \int_C \frac{f(z)}{z - z_0} dz$ $f(z_0) = \frac{1}{2\pi i} \int_C f(z) dz$

Q12

Liouville's theorem states that:

Every bounded entire function is constant Every entire function is constant Every analytic function in a domain is bounded Every bounded function is entire

Q13

Taylor's theorem for complex functions states that if $f$ is analytic in a disk $|z - z_0| < R$, then:

$f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n$ for $|z - z_0| < R$ $f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} z^n$ for $|z| < R$ $f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n$ for $|z - z_0| < R$ $f(z) = \sum_{n=0}^{\infty} a_n z^n$ for $|z| < R$

Q14

Laurent's theorem states that if $f$ is analytic in an annulus $r < |z - z_0| < R$, then:

$f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n$ where $a_n = \frac{1}{2\pi i} \int_C \frac{f(\zeta)}{(\zeta - z_0)^{n+1}} d\zeta$ $f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n$ where $a_n = \frac{f^{(n)}(z_0)}{n!}$ $f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n$ where $a_n = \frac{1}{2\pi i} \int_C f(\zeta) (\zeta - z_0)^{n-1} d\zeta$ $f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n + \sum_{n=1}^{\infty} \frac{b_n}{(z - z_0)^n}$

Q15

If $f$ has a singularity at $z_0$ and $\lim_{z \to z_0} (z - z_0)^n f(z)$ exists and is non-zero for some positive integer $n$, then $z_0$ is:

A pole of order $n$ An essential singularity A removable singularity A branch point

Q16

Riemann's theorem on removable singularities states that if $f$ is analytic and bounded in a punctured neighborhood of $z_0$, then:

$z_0$ is a removable singularity of $f$ $z_0$ is a pole of $f$ $z_0$ is an essential singularity of $f$ $f$ is analytic at $z_0$

Q17

Weierstrass's theorem on essential singularities states that if $z_0$ is an essential singularity of $f$, then:

In every neighborhood of $z_0$, $f$ comes arbitrarily close to every complex value $f$ is unbounded in every neighborhood of $z_0$ $f$ has a Laurent series expansion with infinitely many negative powers Both (a) and (c)

Q18

The residue of a function $f$ at an isolated singularity $z_0$ is:

The coefficient of $(z - z_0)^{-1}$ in the Laurent series expansion of $f$ around $z_0$ The coefficient of $(z - z_0)^0$ in the Laurent series expansion of $f$ around $z_0$ $\frac{1}{2\pi i} \int_C f(z) dz$ where $C$ is a small circle around $z_0$ Both (a) and (c)

Q19

Cauchy's residue theorem states that if $f$ is analytic inside and on a simple closed contour $C$ except for finitely many singularities $z_1, z_2, \ldots, z_n$ inside $C$, then:

$\int_C f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)$ $\int_C f(z) dz = 2\pi \sum_{k=1}^n \text{Res}(f, z_k)$ $\int_C f(z) dz = \sum_{k=1}^n \text{Res}(f, z_k)$ $\int_C f(z) dz = 2\pi i f(z_0)$ where $z_0$ is any point inside $C$

Q20

To evaluate $\int_{-\infty}^{\infty} f(x) dx$ using the method of residues, where $f$ is a rational function with no poles on the real axis and degree of denominator exceeds degree of numerator by at least 2, we:

Consider a semicircular contour in the upper half-plane and evaluate $2\pi i$ times the sum of residues in the upper half-plane Consider a semicircular contour in the lower half-plane and evaluate $2\pi i$ times the sum of residues in the lower half-plane Consider a rectangular contour and evaluate $2\pi i$ times the sum of residues inside the rectangle Consider a keyhole contour and evaluate $2\pi i$ times the sum of residues inside the contour

Q21

The maximum modulus theorem states that if $f$ is analytic and non-constant in a domain $D$, then:

$|f(z)|$ has no maximum in the interior of $D$ $|f(z)|$ has no minimum in the interior of $D$ $f(z)$ has no zeros in the interior of $D$ $f(z)$ is constant in $D$

Q22

The minimum modulus theorem states that if $f$ is analytic and non-zero in a domain $D$, then:

$|f(z)|$ has no minimum in the interior of $D$ unless $f$ is constant $|f(z)|$ has no maximum in the interior of $D$ unless $f$ is constant $f(z)$ has no zeros in the interior of $D$ $f(z)$ is constant in $D$

Q23

Schwarz lemma states that if $f$ is analytic in the unit disk $|z| < 1$, with $f(0) = 0$ and $|f(z)| \leq 1$ for $|z| < 1$, then:

$|f(z)| \leq |z|$ for $|z| < 1$ and $|f'(0)| \leq 1$ $|f(z)| \leq 1$ for $|z| < 1$ and $|f'(0)| \leq 1$ $|f(z)| \leq |z|$ for $|z| < 1$ and $f'(0) = 0$ $|f(z)| \leq 1$ for $|z| < 1$ and $f'(0) = 0$

Q24

A power series $\sum_{n=0}^{\infty} a_n (z - z_0)^n$:

Converges absolutely inside its circle of convergence and diverges outside it Converges uniformly in any compact subset inside its circle of convergence Represents an analytic function inside its circle of convergence All of the above

Q25

The Hadamard formula for the radius of convergence $R$ of a power series $\sum_{n=0}^{\infty} a_n z^n$ is:

$\frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n}$ $R = \limsup_{n \to \infty} |a_n|^{1/n}$ $\frac{1}{R} = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$ $R = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$

Q26

If a power series $\sum_{n=0}^{\infty} a_n (z - z_0)^n$ has radius of convergence $R > 0$, then:

It represents an analytic function in the disk $|z - z_0| < R$ It represents a continuous function in the disk $|z - z_0| < R$ It can be differentiated term by term in the disk $|z - z_0| < R$ Both (a) and (c)

Q27

Rouché's theorem states that if $f$ and $g$ are analytic inside and on a simple closed contour $C$ and $|f(z)| > |g(z)|$ on $C$, then:

$f$ and $f + g$ have the same number of zeros inside $C$ (counting multiplicities) $f$ and $g$ have the same number of zeros inside $C$ (counting multiplicities) $f + g$ has no zeros inside $C$ $f$ has no zeros inside $C$

Q28

The fundamental theorem of algebra states that:

Every non-constant polynomial with complex coefficients has at least one complex root Every polynomial with real coefficients has at least one real root Every polynomial of degree $n$ has exactly $n$ real roots Every polynomial of degree $n$ has exactly $n$ complex roots (counting multiplicities)

Q29

Morera's theorem states that if $f$ is continuous in a domain $D$ and $\int_C f(z) dz = 0$ for every closed contour $C$ in $D$, then:

$f$ is analytic in $D$ $f$ is constant in $D$ $f$ is harmonic in $D$ $f$ is bounded in $D$

Q30

Hurwitz's theorem states that if $(f_n)$ is a sequence of analytic functions in a domain $D$ that converges uniformly to $f$ on compact subsets of $D$, and if $f_n$ has no zeros in $D$ for all $n$, then:

Either $f$ is identically zero in $D$ or $f$ has no zeros in $D$ $f$ has no zeros in $D$ $f$ is identically zero in $D$ $f$ has exactly the same number of zeros as each $f_n$ in $D$

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