MTH2003 Complex Analysis Flashcards

What is the definition of a complex number?card

A complex number is defined as $z=x+iy$, where $i^2=-1$ and $x,y\in \mathbb{R}$. The real part is $\text{Re}(z)=x$, and the imaginary part is $\text{Im}(z)=y$.

How is addition defined for two complex numbers $z_1=x_1+i{y}_1$ and $z_2=x_2+i{y}_2$?card

$z_1+z_2=(x_1+x_2)+i(y_1+y_2)$. This also implies scalar multiplication when one term is real.

What is the formula for multiplying two complex numbers $z_1$ and $z_2$?card

$z_1{z}_2=(x_1{x}_2-y_1{y}_2)+i(x_1{y}_2+x_2{y}_1)$.

How is division performed for complex numbers?card

Multiply numerator and denominator by the conjugate $\overline{z}=x-iy$, then scale by $|z{|}^2$, where $|z|=\sqrt{x^2+y^2}$.

State the Triangle Inequality for complex numbers.card

$|z_1+z_2|\le |z_1|+|z_2|$, and $|z_1{z}_2|=|z_1||z_2|$.

What are the “part separation” properties of complex numbers?card

$z+\overline{z}=2\text{Re}(z)$ and $z-\overline{z}=2i\text{Im}(z)$.

How is a complex number represented in polar form?card

$z=r(\cos \theta +i\sin \theta )$, where $r=|z|$ (modulus) and $\theta =\arg (z)$ (argument).

What is Euler’s Formula, and how does it relate to exponential form?card

$e^{i\theta }=\cos \theta +i\sin \theta$. Thus, $z=r{e}^{i\theta }$ is the exponential form of a complex number.

Solve $Z^3=8i$ using polar form. What are the roots?card

Roots are $2e^{i(\pi /6+2\pi k/3)}$ for $k=0,1,2$: $\sqrt{3}+i$, $-\sqrt{3}+i$, and $-2i$.

What are the $n$-th roots of unity in polar form?card

$Z=e^{i\frac{2\pi k}{n}}$ for $k=0,1,\dots ,n-1$. Example (n=4): $1,i,-1,-i$.

State De Moivre’s Theorem.card

For $z=r(\cos \theta +i\sin \theta )$ and integer $n$, $z^n=r^n(\cos n\theta +i\sin n\theta )$.

How is De Moivre’s Theorem proven using induction?card

Base case $n=1$ holds. Assume true for $n=k$, then show $z^{k+1}=r^{k+1}[\cos (k+1)\theta +i\sin (k+1)\theta ]$ using trigonometric identities. Extend via Euler’s formula.

What is the cyclic group structure of the roots of unity?card

Generated by $\omega =e^{i\frac{2\pi }{n}}$, the group is $\{1,\omega ,{\omega }^2,\dots ,{\omega }^{n-1}\}$, with multiplication corresponding to rotation on the unit circle.

What is Euler’s Identity?card

$e^{i\pi }+1=0$, connecting five fundamental constants in mathematics.

Define the modulus of a complex number $z=x+iy$.card

$|z|=\sqrt{x^2+y^2}$, representing its distance from the origin in the Argand diagram.

What is the conjugate of $z=x+iy$, and how does it interact with $z$?card

$\overline{z}=x-iy$. Properties include $z+\overline{z}=2\text{Re}(z)$ and $z\overline{z}=|z{|}^2$.

What is an open set in the complex plane?card

A subset $D\subset \mathbb{C}$ where every $z_0\in D$ has a neighbourhood $N_{ϵ}(z_0)=\{z:|z-z_0|<ϵ\}$ entirely contained in $D$.

How is a closed set defined in $\mathbb{C}$?card

A set containing all its boundary points, or equivalently, the complement of an open set.

What defines the boundary $\partial S$ of a set $S\subset \mathbb{C}$?card

Points where every neighbourhood contains at least one point in $S$ and one outside $S$.

What is a region in complex analysis?card

An open set with some (or all) of its boundary points added. It is closed if it contains all boundary points.

What is a compact set in $\mathbb{C}$?card

A set that is both closed and bounded (e.g., $\overline{S}=\{z:|z-a|\le r\}$).

Why is $S_5=\{0\}$ a closed set?card

Its complement $\mathbb{C}\setminus \{0\}$ is open, and it contains its only boundary point (itself).

How is a function $f(z)$ of a complex variable expressed in real and imaginary parts?card

$f(z)=u(x,y)+iv(x,y)$, where $z=x+iy$. Example: $f(z)=z^2=(x^2-y^2)+i(2xy)$.

What is the principal value of the complex logarithm?card

$\text{Log}z=\ln |z|+i\text{Arg}z$, where $\text{Arg}z\in (-\pi ,\pi ]$. Example: $\text{Log}i=i\frac{\pi }{2}$.

Why is the complex logarithm multi-valued?card

$e^w=z$ has solutions $w=\ln |z|+i(\arg z+2k\pi )$ for all $k\in \mathbb{Z}$, due to the periodicity of $e^z$.

Name three methods to visualise complex functions.card

1. Parametric plots (separate surfaces for $u(x,y)$ and $v(x,y)$), 2. Domain colouring (colour encodes modulus/argument), 3. Vector fields (plot $(u,v)$ as vectors at each $(x,y)$).

What is the boundary of $S_3=\{z:\text{Re}(z)>0\}$?card

The imaginary axis $\text{Re}(z)=0$.

What is a branch cut? Give an example.card

A curve where a multi-valued function is discontinuous. Example: The negative real axis for $\text{Log}z$.

Express $e^z$ in terms of real and imaginary parts.card

$e^z=e^x\cos y+i{e}^x\sin y$, where $z=x+iy$.

What is the domain of $g(z)=\frac{1}{z+i}$?card

$\mathbb{C}\setminus \{-i\}$ (all complex numbers except $-i$).

Classify $S_4=\{z:y>x^2\}$.card

An open set; its boundary is the parabola $y=x^2$.

How is the closure $\overline{S}$ of an open disk $S$ defined?card

$\overline{S}=S\cup \partial S=\{z:|z-a|\le r\}$.

What is a neighbourhood of $z_0\in \mathbb{C}$?card

An open disk $N_{ϵ}(z_0)=\{z:|z-z_0|<ϵ\}$.

Solve $\ln i$. What are all its values?card

$\ln i=i\left(\frac{\pi }{2}+2k\pi \right)$ for $k\in \mathbb{Z}$. Principal value: $i\frac{\pi }{2}$.

What distinguishes a bounded region in $\mathbb{C}$?card

It can be enclosed in a finite disk (e.g., $|z|<5$).

Give an example of a closed set that is not compact.card

The upper half-plane $\text{Im}(z)\ge 1$ is closed but unbounded, hence not compact.

Why are multi-valued functions like $\ln z$ not true functions?card

They assign multiple outputs to a single input. Restricting the domain (e.g., using a branch) makes them single-valued.