Complex Analysis

Syllabus: Analytic functions, Cauchy's integral theorem, Cauchy's integral formula, sequences, series, convergence tests, Taylor and Laurent series, residue theorem.

8 hand-picked visual resources on this topic, 6 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Analytic functions and C–R equations

A function is analytic where it is differentiable in a neighbourhood. $e^z$ , $\sin z$ , $\cos z$ and polynomials are entire; anything with a pole, a branch cut or a conjugate is not.

$u_x=v_y,\qquad u_y=-v_x$

Where marks are lost

$1/(1-z)$ and $e^{1/z}$ are not entire — they fail at $z=1$ and

See it

Residue theorem

Only the poles inside the contour contribute. Locate the poles, test each against the contour, then sum residues.

$\oint_C f(z)\,dz=2\pi j\sum \text{Res}$

Cauchy's integral theorem and formula

If $f$ is analytic everywhere inside and on a closed contour, the integral around it is zero — so the only thing that ever makes a contour integral non-zero is a singularity inside it. The integral formula then says the value at a point is fixed by the values around it.

$\oint_C f(z)\,dz=0$

Taylor, Laurent and convergence

A Taylor series exists where $f$ is analytic; a Laurent series is what you get around a singularity, and it carries negative powers. The coefficient of $(z-z_0)^{-1}$ in it is

Below: on the syllabus, but no past question for it in this bank.

Roots of a complex number
syllabus

Write the number in polar form; the $n$ roots are equally spaced by $2\pi/n$ around a circle of radius $|w|^{1/n}$ .

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Complex Analysis

Syllabus: Analytic functions, Cauchy's integral theorem, Cauchy's integral formula, sequences, series, convergence tests, Taylor and Laurent series, residue theorem.

8 hand-picked visual resources on this topic, 6 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Analytic functions and C–R equations

A function is analytic where it is differentiable in a neighbourhood. $e^z$ , $\sin z$ , $\cos z$ and polynomials are entire; anything with a pole, a branch cut or a conjugate is not.

$u_x=v_y,\qquad u_y=-v_x$

Where marks are lost

$1/(1-z)$ and $e^{1/z}$ are not entire — they fail at $z=1$ and

See it

Residue theorem

Only the poles inside the contour contribute. Locate the poles, test each against the contour, then sum residues.

$\oint_C f(z)\,dz=2\pi j\sum \text{Res}$

Cauchy's integral theorem and formula

$\oint_C f(z)\,dz=0$

Taylor, Laurent and convergence

A Taylor series exists where $f$ is analytic; a Laurent series is what you get around a singularity, and it carries negative powers. The coefficient of $(z-z_0)^{-1}$ in it is

Below: on the syllabus, but no past question for it in this bank.

Roots of a complex number
syllabus

Write the number in polar form; the $n$ roots are equally spaced by $2\pi/n$ around a circle of radius $|w|^{1/n}$ .

Complex Analysis

Analytic functions and C–R equations

Residue theorem

Cauchy's integral theorem and formula

Taylor, Laurent and convergence

Roots of a complex numbersyllabus

Complex Analysis

Analytic functions and C–R equations

Residue theorem

Cauchy's integral theorem and formula

Taylor, Laurent and convergence

Roots of a complex numbersyllabus

Roots of a complex number
syllabus

Roots of a complex number
syllabus