Complex Analysis Review

Everything from the section, arranged for lookup: the definitions, the named theorems, the standard maps, the contour recipes, and the notation. Use it as a reference rather than reading straight through.

Arithmetic in polar form is the working skill:

$$z_1z_2 = r_1r_2e^{i(\theta_1+\theta_2)}, \qquad z^n = r^ne^{in\theta}$$

Multiply the moduli, add the arguments. Multiplication by $i$ is a quarter turn.

• $|z_1z_2| = |z_1||z_2|$, but $|z_1+z_2|$ has no formula, only the triangle inequality.
• $z\bar z = |z|^2$, which is how division is done.
• $z^n = w$ has exactly $n$ solutions, forming a regular $n$-gon.
• The $n$th roots of unity sum to 0 and form a cyclic group of order $n$.

Set descriptions: $|z-a|=r$ is a circle, $|z-a|<r$ a disc, $r_1<|z-a|<r_2$ an annulus, $\alpha<\operatorname{Arg}z<\beta$ a sector.

Complex differentiability requires the difference quotient’s limit to be the same from every direction. That forces the Cauchy–Riemann equations, and geometrically it forces the map to be locally a rotation and a uniform scaling.

Fails everywhere: $\bar z$, $|z|$, $\operatorname{Re}z$, $\operatorname{Im}z$. Fails except at one point, hence analytic nowhere: $|z|^2$.

Analytic means differentiable on an open set, not at a point. Everything below needs the open set.

• Infinite differentiability, and each derivative analytic
• Equality with the Taylor series on any disc in the domain
• The identity theorem: agreement on a set with a limit point forces global agreement
• The maximum modulus principle: no interior maximum of $|f|$
• Liouville: bounded plus entire implies constant
• Open mapping, and conformality where $f'\ne0$
• $u$ and $v$ harmonic, with orthogonal level curves

$$e^z = e^x(\cos y+i\sin y), \qquad e^{z+2\pi i}=e^z, \qquad e^z \ne 0$$

Periodicity is the source of everything awkward. It makes $\log$ multi-valued:

$$\log z = \ln|z|+i(\operatorname{Arg}z+2\pi k), \qquad \operatorname{Log}(-1)=i\pi$$

which needs a branch cut, conventionally the negative real axis. $\operatorname{Log}(z_1z_2) = \operatorname{Log}z_1+\operatorname{Log}z_2$ can fail by $2\pi i$.

$\sin z$ and $\cos z$ are entire and unbounded, since $\sin(iy)=i\sinh y$; Liouville requires it. Their zeros stay real. Powers $z^a = e^{a\log z}$ give one value for integer $a$, $q$ values for $a=p/q$, and infinitely many otherwise; $i^i = e^{-\pi/2-2\pi k}$ is always real.

$$\int_\gamma f\,dz = \int_a^b f(\gamma(t))\gamma'(t)\,dt$$

The one computation to memorize:

$$\oint_{|z|=1}\frac{dz}{z} = 2\pi i, \qquad \oint_{|z|=1}z^n\,dz = 0 \ (n\ne-1)$$

The exponent $-1$ being the sole exception is why residues exist.

ML inequality: $\left|\int_\gamma f\right|\le ML$. Used to kill large arcs.

Theorem	Statement
Cauchy’s theorem	$f$ analytic on a simply connected domain $\Rightarrow \oint_C f = 0$
Deformation	Reshaping a contour changes nothing unless it crosses a singularity
Multiply connected	An outer contour equals the sum of contours around the holes
Integral formula	$f(z_0) = \frac{1}{2\pi i}\oint_C\frac{f(z)}{z-z_0}dz$
Derivative formula	$f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_C\frac{f(z)}{(z-z_0)^{n+1}}dz$
Mean value property	$f(z_0)$ is the average of $f$ over any circle about it
Morera	Vanishing closed integrals plus continuity implies analytic

Cauchy’s theorem is the Cauchy–Riemann equations integrated, via Green’s theorem. The integral formula is where infinite differentiability comes from, and it gives Liouville, then the fundamental theorem of algebra, in a few lines each.

Taylor. Analytic on a disc means equal to the Taylor series there. Over $\mathbb{C}$, analytic and locally a power series coincide, unlike over $\mathbb{R}$.

$$R = \text{distance from the centre to the nearest singularity}$$

Laurent. On an annulus $r<|z-z_0|<R$,

$$f(z) = \sum_{n=-\infty}^{\infty}c_n(z-z_0)^n$$

unique for that annulus. One function can have several Laurent series about the same centre, one per annulus of analyticity. Compute them with partial fractions plus geometric series, deciding which factor to pull out by which ratio has modulus below 1.

The consequence that matters:

$$\oint_C f\,dz = 2\pi i\,c_{-1}$$

Type	Limit behaviour	Principal part
Removable	$\lim f$ exists (boundedness suffices)	empty
Pole of order $m$	$\lvert f\rvert\to\infty$	$m$ terms
Essential	no limit at all	infinite

For $f=p/q$ with zero orders $a$ and $b$: removable if $a\ge b$, pole of order $b-a$ if $a<b$. Cancel before classifying.

Near an essential singularity, Casorati–Weierstrass gives a dense image and Picard gives every value infinitely often with at most one exception.

Residue formulas:

$$\text{simple: } \lim_{z\to z_0}(z-z_0)f(z), \qquad \frac{p(z_0)}{q'(z_0)}$$ $$\text{order } m: \ \frac{1}{(m-1)!}\lim_{z\to z_0}\frac{d^{m-1}}{dz^{m-1}}\big[(z-z_0)^mf(z)\big]$$

Essential singularities have no formula; expand the series.

Residue theorem: $\oint_C f = 2\pi i\sum\operatorname{Res}$ over enclosed singularities only.

Integral	Contour	Extra tool
$\int_{-\infty}^\infty \frac{p}{q}$, $\deg q\ge\deg p+2$	upper semicircle	ML on the arc
$\int_{-\infty}^\infty f(x)e^{iax}dx$, $a>0$	upper semicircle	Jordan’s lemma
$\int_0^{2\pi}R(\cos\theta,\sin\theta)d\theta$	unit circle, $z=e^{i\theta}$	$d\theta = dz/(iz)$
$\int_0^\infty$ with $x^n$ symmetry	sector of angle $2\pi/n$	match the symmetry
$\int_0^\infty x^{a-1}f(x)dx$	keyhole	the branch jump
pole on the real axis	indented contour	half-loop gives $\pi i\operatorname{Res}$

Reference values: $\int\frac{dx}{1+x^2}=\pi$, $\int\frac{dx}{1+x^4}=\frac{\pi}{\sqrt2}$, $\int\frac{\cos x}{1+x^2}dx=\frac{\pi}{e}$, $\int_0^\infty\frac{\sin x}{x}dx=\frac{\pi}{2}$.

Conformal means angle-preserving, and an analytic $f$ is conformal exactly where $f'\ne0$. At a zero of order $m$ in $f'$, angles are multiplied by $m+1$.

Möbius maps $\frac{az+b}{cz+d}$ with $ad-bc\ne0$ are bijections of $\hat{\mathbb{C}}$, form a group, send circles and lines to circles and lines, and are determined by three points.

Map	Effect
$az+b$	similarity
$1/z$	inversion
$\frac{z-i}{z+i}$	upper half plane $\to$ disc
$z^2$	quarter plane $\to$ half plane
$e^z$	strip of height $\pi$ $\to$ half plane
$z+\frac1z$	circle $\to$ segment (Joukowski)

Riemann mapping theorem: every simply connected domain other than $\mathbb{C}$ maps conformally onto the disc. Existence only, no formula.

Argument principle: $\frac{1}{2\pi i}\oint\frac{f'}{f} = Z-P$, equal to the winding number of the image around 0.

Rouché: $|g|<|f|$ on $C$ implies $f$ and $f+g$ have the same zero count inside. Gives the fundamental theorem of algebra with the exact count.

$\nabla^2u=0$. The real and imaginary parts of an analytic function are harmonic, and on a simply connected domain every harmonic function has a harmonic conjugate, unique up to a constant. On an annulus $\ln|z|$ shows the conjugate can fail to exist globally.

Inherited properties: mean value, maximum principle, uniqueness from boundary data, infinite smoothness, Liouville.

The Dirichlet problem is solved on the disc by the Poisson integral formula and elsewhere by conformal mapping. Level curves of $u$ and $v$ are the equipotentials and field lines, or isotherms and heat flow, or velocity potential and streamlines.

Theorem	Statement
Cauchy–Riemann	$u_x=v_y$ and $u_y=-v_x$ characterize differentiability
Cauchy’s theorem	$\oint_C f = 0$ on a simply connected domain
Cauchy’s integral formula	Boundary values determine interior values
Morera	Converse of Cauchy’s theorem
Liouville	Bounded plus entire implies constant
Fundamental theorem of algebra	Every non-constant polynomial has a root
Maximum modulus	No interior maximum of $\lvert f\rvert$
Identity theorem	Agreement on a set with a limit point forces global agreement
Taylor	Analytic on a disc means equal to the power series
Laurent	Analytic on an annulus means a two-sided series
Riemann (removable)	Bounded near a singularity implies removable
Casorati–Weierstrass	Dense image near an essential singularity
Picard	Every value, infinitely often, at most one exception
Residue theorem	$\oint_C f = 2\pi i\sum\operatorname{Res}$
Argument principle	$\frac{1}{2\pi i}\oint\frac{f'}{f}=Z-P$
Rouché	$\lvert g\rvert<\lvert f\rvert$ preserves the zero count
Riemann mapping	Simply connected, not $\mathbb{C}$, maps onto the disc
Jordan’s lemma	Oscillatory arcs vanish with only $f\to0$

• Forgetting simple connectivity in Cauchy’s theorem. $\oint\frac{dz}{z}=2\pi i$ is the standing counterexample.
• Confusing $\arg$ with $\operatorname{Arg}$, or ignoring the quadrant when computing an argument.
• Applying $\operatorname{Log}(z_1z_2)=\operatorname{Log}z_1+\operatorname{Log}z_2$ to principal values.
• Assuming $\lvert\sin z\rvert\le1$. True on $\mathbb{R}$ only.
• Saying “differentiable at a point” when analyticity is needed. $\lvert z\rvert^2$ is the reminder.
• Not cancelling before classifying a singularity. $\frac{\sin z}{z^3}$ has a pole of order 2, not 3.
• Assuming residue and order are linked. A high-order pole can have residue 0.
• Including residues of poles outside the contour.
• Reading a Laurent series as belonging to a function alone. It belongs to a function and an annulus.
• Forgetting to check the arc vanishes before equating a real integral with $2\pi i\sum\operatorname{Res}$.
• Using a conformal map through a critical point, where $f'=0$ and angles are not preserved.
• Expecting a harmonic conjugate on a region with a hole.