Cauchy's integral theorem and formula: Foundations

Before you can read the parent note, you need to see what every symbol means. We build them in order, so each new one only uses ones already made.

1. $C$ — the complex plane (the stage everything lives on)

Plain words: think of $z$ as an address on a flat map. You walk $x$ steps east and $y$ steps north. That landing spot is the number. The set $C$ is simply "all possible addresses on the map."

The picture: every complex number is a dot in a plane. The horizontal axis holds the "real part" $x$ (written $Re z$ ), the vertical axis holds the "imaginary part" $y$ (written $Im z$ ).

Figure — Cauchy's integral theorem and formula

Figure s01 — the complex plane: an amber arrow runs from the origin to the dot $z = x + i y$ ; a dashed cyan line drops to the horizontal axis marking $x = Re z$ , another runs left to the vertical axis marking $y = Im z$ .

Why the topic needs it: Cauchy's theorem integrates over a path in this plane. If $z$ were just a number on a line, there would be no loops to walk around and nothing to prove. The whole subject exists because $C$ is two-dimensional.

2. $e$ , the exponential, and $e^{i θ}$ — before we use them

We define $∣ z ∣$ and $θ$ properly in the next section; here we only earned the symbol $e^{i θ}$ so it can be used later.

3. $∣ z ∣$ and $θ$ — size and direction (polar view)

The picture: draw the arrow from the origin $0$ to the dot $z$ . Its length is $∣ z ∣$ ; its tilt is $θ$ . That is the same right-triangle idea used for real vectors — $x$ is the adjacent side, $y$ the opposite side.

Figure s02 — the unit circle in cyan; an amber arrow of length $1$ at angle $θ$ lands on the point $e^{i θ} = cos θ + i sin θ$ . A small white arc at the origin marks the angle $θ$ .

Why the topic needs it: the integral formula parametrises a tiny circle as $z = a + ε e^{i θ}$ . That notation is pure polar: $ε$ is the modulus (radius) and $θ$ sweeps the argument all the way around. Without polar form you cannot even write the key step.

4. A function $f : C \to C$ — a rule that moves points

Plain words: feed in one dot, get out another dot. The whole plane gets picked up, stretched, twisted, and set back down.

Why the topic needs it: the parent writes $f = u + i v$ and $d z = d x + i d y$ , then splits the loop integral into a real part and an imaginary part. That split is this $u, v$ decomposition combined with $d z = d x + i d y$ . You cannot follow the Green's-theorem step without both.

5. The derivative as a limit — and why direction matters

The picture: stand at a point $z_{0}$ . Take a tiny step $h$ to a nearby point. Ask: how much did the output move, per unit of input step? That ratio $\frac{f ( z _{0} + h ) - f ( z _{0} )}{h}$ is the local stretch-and-turn factor.

Figure s03 — the point $z_{0}$ (amber dot) with eight cyan arrows $h$ fanning out in all directions; the caption of the figure stresses that the limit must give one and the same value along every one of them.

Why the topic needs it — and why this tool: In real calculus a derivative only checks left and right (two directions). Here $h$ lives in a plane, so there are infinitely many approach directions. Demanding one answer for all of them is a colossal constraint. That constraint is what forces the Cauchy–Riemann equations, which are what make loop integrals vanish. We use a limit because "instantaneous stretch factor" has no meaning without shrinking the step to zero.

Intuition The two directions that create Cauchy–Riemann

Since every direction must give the same $f^{'} (z_{0})$ , in particular the horizontal approach and the vertical approach must agree. Watching those two chosen directions is what pins down the equations in the next section.

Horizontal approach — let $h = Δ x$ be a tiny real step (pure east). Only $x$ changes, so $f^{'} (z_{0}) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} .$ Vertical approach — let $h = i Δ y$ be a tiny imaginary step (pure north). Now $h$ has a $\frac{1}{i} = - i$ factor out front, giving $f^{'} (z_{0}) = \frac{1}{i} (\frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y}) = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y} .$ Setting these two equal — real part to real part, imaginary part to imaginary part — is exactly what forces the Cauchy–Riemann equations in §6. (The symbols $\partial u / \partial x$ etc. are defined right below; read §6 first if they are new to you, then return.)

6. Partial derivatives $\partial u / \partial x$ — slope in one chosen direction

The picture: $u (x, y)$ is a height landscape over the plane. Walk due east — how steeply does the ground rise? That is $\partial u / \partial x$ . Walk due north — that is $\partial u / \partial y$ .

Why the topic needs it: the horizontal approach in §5 uses $\partial_{x}$ ; the vertical approach uses $\partial_{y}$ . Matching the two directional limits term-by-term produces the Cauchy–Riemann equations $u_{x} = v_{y}, u_{y} = - v_{x}$ (the first pairs the real parts, the second the imaginary parts). Partial derivatives are the language those equations are written in.

7. $γ$ , the contour — a path you walk

The picture: a bent, possibly looping arrow-path drawn on the complex plane. The little arrowhead shows travel direction.

Why the topic needs it: $\oint_{γ}$ means "add up along the loop $γ$ ." The circle symbol on the integral sign, $\oint$ , specifically means the path closes on itself. Everything Cauchy proves is about these loops.

8. $\oint_{γ} f (z) d z$ — the contour integral itself

The picture: at every point on the path, an arrow $f (z)$ tells you a local push; $d z$ is the direction you step. Their product is one contribution; the integral is the grand total of all contributions around the loop.

Figure s04 — a cyan closed loop $γ$ with a direction arrowhead (counter-clockwise); six amber arrows tangent to the loop mark the tiny steps $d z$ , one at each sample point, illustrating "sum $f (z)$ times each $d z$ around the loop."

Why the topic needs it: this is the object Cauchy's theorem sets to zero. Understanding it as "weighted sum of tiny steps" is what makes "the total cancels to $0$ " believable rather than magic.

9. Singularities, regions, and "simply connected"

The picture: a solid pancake (fine) versus a pancake with a bite missing (a loop hugging the bite is trapped).

Why the topic needs it: the Green's-theorem step demands $f$ be holomorphic on the entire filled interior of $γ$ . A single singularity inside is a "hole" that breaks simple connectivity, and the proof collapses there. This is exactly why $\oint d z / z$ is not zero — the interior has a singularity at the origin.

The prerequisite map (how to read it)

Each box is a foundation. Read every arrow as the sentence "you need the box at the tail before the box at the head makes sense." Follow the arrows from the top boxes down; everything eventually feeds the bottom box — the parent topic.

Every arrow says "you need the left box before the right box makes sense." The three streams — the algebra of $C$ , the calculus of $f$ , and the geometry of paths — all merge at the parent topic Cauchy's integral theorem and formula.

Why these feed forward (one line each)

Cauchy-Riemann equations are born from §5 (the two directional limits) + §6 (partials).
Green's theorem is what converts the loop integral of §8 into an area integral that the CR equations kill.
Once loops vanish, deforming a contour to a tiny circle gives the integral formula, and repeating the trick with higher powers of $(z - a)$ gives the Residue theorem.
The rigidity you glimpse in §5 later powers Liouville's theorem and the Maximum modulus principle.

Equipment checklist

Cover the right side and test yourself.

What set does the symbol

C

denote?

C = {x + i y ∣ x, y \in R}

— the field of all complex numbers.

What is

i^{2}

, and what geometric action does multiplying by

i

perform?

i^{2} = - 1

; multiplying by

i

rotates a point 90° counter-clockwise.

What are

Re z

and

Im z

for

z = x + i y

Re z = x

(horizontal part),

Im z = y

(vertical part).

What is

e

and what defines the exponential

e^{x}

e \approx 2.71828

;

e^{x}

is the function equal to its own slope, built by the series

1 + x + x^{2} /2! + \dots

State Euler's form and what

e^{i θ}

is geometrically.

e^{i θ} = cos θ + i sin θ

; it is the point on the unit circle at angle

θ

Why is the argument

θ

multi-valued, and how do we pin it down?

Adding any whole turn

2 π

lands on the same point; we fix the principal value

- π < θ \leq π

using a branch cut.

Write the differential

d z

in terms of

d x

and

d y

d z = d x + i d y

— a tiny east step plus

i

times a tiny north step.

f (z) = u + i v

, what are

u

and

v

Real functions of

x, y

u

is the real part of the output,

v

the imaginary part.

Why is the complex derivative stricter than the real one?

The limit must give the same value from every direction in the plane, not just left/right.

Which two approach directions produce the Cauchy–Riemann equations?

The horizontal (

h = Δ x

) and vertical (

h = i Δ y

) approaches, set equal.

What does

\partial u / \partial x

measure?

Slope of

u

when only

x

is nudged and

y

is held fixed.

What does the circle on

\oint_{γ}

signify?

The contour

γ

is closed — it ends where it began (a loop).

In words, what does

\oint_{γ} f d z

compute?

The sum of

f (z)

times each tiny path-step

d z

all the way around the loop.

What makes a point a singularity of

f

f

fails to be holomorphic there — typically

f (z)

blows up, e.g.

1/ z

z = 0

What is a region (domain) in complex analysis?

A set that is open (no boundary points) and connected (any two points joinable by an interior path).

What does "simply connected" mean?

Every loop in the region can be shrunk to a point without leaving it — no holes.

Why does one enclosed singularity break Cauchy's theorem?

It punches a hole, so the interior is no longer simply connected and Green's theorem cannot be applied over the whole region.