Problem-solving recipes

A step-by-step procedure for every problem type this course teaches — what to do first, what to do next, and how to tell which case you're in.

What are you being asked to do?

Start with what you're actually being asked to produce.

Convert a complex number between rectangular and exponential form — the readiness recipe.

Factor a polynomial completely over $\mathbb C$ , or count its roots — the fundamental-theorem-of-algebra recipe.

Given one harmonic function, find its harmonic conjugate — the Cauchy-Riemann recipe.

Evaluate $\exp(z)$ or $\operatorname{Log}(z)$ for a specific $z$ — the two elementary-function recipes; the logarithm one is also where branch-cut subtleties like $\operatorname{Log}(z^2)$ vs. $2\operatorname{Log}(z)$ get resolved.

Evaluate a contour integral — parametrize and integrate directly, unless an antiderivative exists on the whole domain, in which case the antiderivative recipe is faster; unless the integrand is a quotient with a power in the denominator, in which case the Cauchy derivative-formula recipe applies instead of either.

Need a Laurent series on one specific named annulus, not just anywhere it happens to converge — the Laurent-in-an-annulus recipe.

Given an isolated singularity, need to know removable, pole, or essential — the singularity-classification recipe; once it's confirmed to be a pole, the residue-at-a-pole recipe finds $\operatorname{Res}_{z=z_0}f$ from that classification.

Need a real improper integral over the whole line, or a real integral of sine and cosine over one period — the semicircular- contour recipe for the first shape, the unit-circle-substitution recipe for the second.

Given two points in the unit disk, need a pseudo-hyperbolic distance or an automorphism-recognition verdict — the Schwarz-Pick recipe; given three points and where each one should land instead, the three-point Mobius-construction recipe.

Evaluate $\Gamma(z)$ — the defining-integral recipe for $\operatorname{Re}(z)>0$ directly; the functional-equation recipe for values reached by recursion or the reflection formula instead.

Readiness

Convert between rectangular and exponential form

Use when: you have $z=a+bi$ and need $z=re^{i\theta}$ , or you have $z=re^{i\theta}$ and need $a+bi$

Goal: the same complex number rewritten in whichever of the two forms you didn't start with, with $r>0$ and $\theta$ picked correctly by quadrant

Polar and exponential form, revisited

Factor a polynomial completely over $\mathbb C$

Use when: you're given a polynomial of degree $n\ge3$ and at least one of its roots, and asked for the complete factorization into $n$ linear factors

Goal: the polynomial written as a product of exactly $n$ linear factors, with every root the Fundamental Theorem promises actually accounted for

The Fundamental Theorem of Algebra, revisited

Harmonic functions

Find a harmonic conjugate of a harmonic function

Use when: you have one harmonic function, $u(x,y)$ or $v(x,y)$ , and need the other half of an analytic $f=u+iv$

Goal: the harmonic conjugate, correct up to an added real constant, built directly from the Cauchy-Riemann equations

Harmonic conjugates

Elementary complex functions

Evaluate the complex exponential $\exp(z)$

Use when: you're given $z=x+iy$ and need $\exp(z)$ written out in $a+bi$ form

Goal: the value of $\exp(z)$ as an explicit real and imaginary part, with any full lap already accounted for in $y$

The complex exponential function

Evaluate the principal logarithm $\operatorname{Log}z$

Use when: you need $\operatorname{Log}z$ for a specific $z\neq0$ , or you're comparing $\operatorname{Log}(z^n)$ against $n\operatorname{Log}z$

Goal: the single principal value $\ln|z|+i\operatorname{Arg}z$ , with $\operatorname{Arg}z$ correctly picked from $(-\pi,\pi]$

The logarithm and its branches

Complex integration

Evaluate a contour integral

Use when: you're given $\int_C f(z)\,dz$ for a specific contour $C$ , stated as an explicit parametrization or as a curve you can parametrize yourself (a line segment, a circular arc, a bent path with corners)

Goal: the contour integral's value, as a single complex number, by rewriting it as an ordinary integral over the real parameter $t$

Contour integrals

Evaluate a contour integral using an antiderivative

Use when: you're asked for $\int_C f(z)\,dz$ and you can write down a candidate $F$ with $F'(z)=f(z)$

Goal: the contour integral's value as $F(z_1)-F(z_0)$ , in one step, with no parametrization or integration along $C$ required

Antiderivatives and path independence

Faster than direct parametrization when an antiderivative exists on the whole domain.

Evaluate a contour integral with the derivative formula

Use when: the integrand has a single point $z_0$ raised to a power in the denominator, $\dfrac{f(z)}{(z-z_0)^{k}}$ , with $f$ analytic and $z_0$ strictly inside a simple closed contour $C$

Goal: the integral's exact value, obtained from a derivative of $f$ at one point instead of any integration

The derivative formula

For a quotient with a power in the denominator, once Cauchy's theorem applies.

Laurent series and singularities

Expand a function as a Laurent series in a given annulus

Use when: you're asked for the Laurent series of $f$ centered at $z_0$ , valid specifically on one named annulus between two of $f$ 's singularities

Goal: the two-sided series $\sum c_n(z-z_0)^n$ that actually converges to $f$ on that one annulus, not some other one centered at the same point

Laurent's theorem

Classify an isolated singularity

Use when: you have an isolated singularity of $f$ at $z_0$ and need to know whether it is removable, a pole, or essential

Goal: a firm classification of $z_0$ , plus, when it's a pole, its exact finite order $m$

Isolated singularities -- removable, pole, and essential

Find the residue at a pole of finite order

Use when: you've confirmed $z_0$ is a pole of finite order $m$ and need $\operatorname{Res}_{z=z_0}f$

Goal: the residue $\operatorname{Res}_{z=z_0}f$ , computed from a finite number of derivatives instead of a full Laurent expansion

The pole order and residue shortcut formula

Needs the classification recipe above first, to confirm z0 is a pole and find its order.

Applications of residues

Evaluate a real improper integral using a semicircular contour

Use when: you need $\int_{-\infty}^\infty f(x)\,dx$ for a real rational function $f=p/q$ with real coefficients, no pole on the real axis, and $\deg q\ge\deg p+2$

Goal: the improper integral's exact value, built entirely from the poles in the upper half-plane, with no antiderivative required

Real improper integrals via semicircular contours

Evaluate a trigonometric integral using the unit circle

Use when: you need $\int_0^{2\pi}R(\cos\theta,\sin\theta)\,d\theta$ for $R$ a rational function of sine and cosine, over one full period, with no singularity on the circle itself

Goal: the trigonometric integral's exact value, rewritten as a contour integral around $|z|=1$ and finished by residues

Trigonometric integrals via the unit circle

Mobius transformations and conformal mapping

Find a Mobius transformation sending three points to three points

Use when: you're given three distinct points $z_1,z_2,z_3$ and three distinct images $w_1,w_2,w_3$ , and asked for the one Mobius transformation $T$ with $T(z_k)=w_k$

Goal: the explicit transformation $T(z)=\dfrac{az+b}{cz+d}$ satisfying all three point conditions at once

Möbius transformations

Compute pseudo-hyperbolic distance and read what it implies

Use when: you have a point $a\in\mathbb D$ to recenter to $0$ with $\varphi_a$ , two points needing a pseudo-hyperbolic distance $\rho(z_1,z_2)$ , or an equality $\rho(f(z_1),f(z_2))=\rho(z_1,z_2)$ to interpret

Goal: a confirmed $\varphi_a(a)=0$ , a distance between $0$ and $1$ measuring how far apart two points sit inside $\mathbb D$ , and, when equality holds, the conclusion that $f$ is a disk automorphism everywhere

The Schwarz-Pick theorem

The gamma function

Evaluate Gamma(z) from the defining integral

Use when: you're asked for $\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt$ at a specific $z$ , using the integral itself rather than a known identity

Goal: either the integral's exact value at that $z$ , or a confirmation that it genuinely diverges there

The Gamma function

Evaluate Gamma(z) using the functional equation

Use when: you need $\Gamma(z)$ at an integer, a half-integer, or any point reached from a known value by whole-integer shifts, especially where $\operatorname{Re}(z)\le0$ so the defining integral doesn't converge

Goal: $\Gamma(z)$ 's value, or confirmation that $z$ is a pole with no finite value, built from a known starting value using only algebra, no integration

The functional equation and the reflection formula

A genuinely different technique (algebraic recursion and reflection) from the defining-integral recipe above, not an extension of it.