Intuition The ONE core ideaA contour integral is nothing but adding up little arrows as you walk along a path in a flat 2D plane . Everything else — Cauchy's theorem, residues, 2 π i when you walk all the way around a loop, do the arrows cancel to zero, or does a "spike" inside force a leftover?
Before you can read the parent note you must own nine building blocks . We install them one at a time, each one earning its place before the next. Never do we use a symbol we have not first drawn.
Definition A complex number is a point in a planeWrite z = x + i y x right you go, y up . The symbol i z address ( x , y ) complex plane and written C
Look at the figure. The horizontal axis is the real axis (all the ordinary numbers). The vertical axis is the imaginary axis . The point z = 3 + 2 i 3 2 all i
Intuition Why a plane and not a line?Ordinary calculus lives on a line: you can only go left or right. Complex integration needs two directions of freedom because a path can curve, loop, and enclose regions. The whole subject exists because z
x z Re ( z ) the horizontal coordinate.
y Im ( z ) the vertical coordinate (the number multiplying i
Definition Modulus = length of the arrowDraw an arrow from the origin 0 z length is
∣ z ∣ = x 2 + y 2 . x y
Why do we need it? The parent note says things like "± i ∣ z ∣ = 2 ∣ ± i ∣ = 1 < 2 only readable if you know ∣ z ∣ ∣ z ∣ = 2 2
∣3 + 2 i ∣ = 9 + 4 = 13 ≈ 3.6 ∣ i ∣ = 0 2 + 1 2 = 1 i
Definition Euler's spinnerThe symbol e i t the point on the unit circle at angle t :
e i t = cos t + i sin t . t 0 2 π counterclockwise once around the unit circle , starting at 1 0 1 2 π
In the figure, t t = 0 1 + 0 i t = 2 π i t = π − 1 t = 2 3 π − i
Intuition Why this and not "x 2 + y 2 = 1 x 2 + y 2 = 1 describes the circle but gives no clock — no sense of which way or how fast you move. Contour integration is a walk , so we need a moving point with a time variable. e i t every example in the parent uses z ( t ) = e i t
e i t t ∣ e i t ∣ = 1 on the unit circle , never runs off to infinity. The exponential of a real number grows; the exponential of an imaginary number rotates.
Definition Contour and parametrisationA contour γ gamma ) is a curve in the plane — a road. To drive it we give a parametrisation z ( t ) t ∈ [ a , b ] t t = a t = b
The figure shows two contours: a straight segment z ( t ) = t ( 1 + i ) 0 1 + i circular loop z ( t ) = e i t ∮ ∫ closed loop — you end where you started.
Intuition Why "piecewise-smooth"?"Smooth" means the road has a well-defined direction of travel (a velocity) at every point. "Piecewise" allows a few sharp corners — like a square contour — so long as each straight bit is smooth. We need direction because, as we'll see, the direction of travel is part of the answer.
Symbol γ the path/contour you integrate along.
Symbol ∮ an integral over a closed loop.
z ( t ) your position at clock-time t
d z derivative z ′ ( t ) velocity — a vector telling you which way and how fast you're moving at time t d z = z ′ ( t ) d t tiny step-arrow : a small displacement along the road, carrying direction .
Why does direction matter? In ordinary calculus d x d z complex number , so it has both a length and an angle. When we sum ∑ f ( z k ) Δ z k f ( z k ) This is why z ′ ( t ) — it's not decoration, it is the direction of the walk.
Worked example Velocity on the unit circleIf z ( t ) = e i t z ′ ( t ) = i e i t i 90° perpendicular to your position arrow — exactly what "going around a circle" means. In the parent's Step 1 this i e i t 2 π i
Common mistake Dropping thez ′ ( t ) ∮ z d z = ∫ 0 2 π e i t 1 d t i e i t wrong answer. Always turn d z z ′ ( t ) d t
Definition Splitting a complex function into two real onesA complex function f z = x + i y f ( z ) = u ( x , y ) + i v ( x , y ) . u v ordinary real functions of two real variables . Think of f f ( z ) z
This split is exactly what lets the parent note apply real-calculus tools (Green's theorem ) to a complex problem: it turns one complex integral into two real ones.
f ( z ) = z 2 z 2 = ( x + i y ) 2 = ( x 2 − y 2 ) + i ( 2 x y ) u = x 2 − y 2 v = 2 x y
u the real part of the output f ( z )
v the imaginary part of the output f ( z )
Definition Analytic functionf analytic at a point if it has a genuine complex derivative there — meaning the rate of change h f ( z + h ) − f ( z ) one answer no matter which direction the tiny step h much stronger demand than real differentiability.
Analytic = the arrow-field f ( z ) 0 Cauchy–Riemann equations u x = v y , u y = − v x
z ˉ z ˉ = x − i y h not analytic. That is why ∫ γ z ˉ d z
Definition Singularity / poleA singularity of f f pole is a specific kind: where f ( z − z 0 ) k something nonzero z 0 z 1 0
The residue of f z 0 Res z 0 f z − z 0 1 Laurent series (a power series that allows negative powers). The parent's Step 2 shows only this 1/ z only thing a closed-loop integral can ever detect.
For a simple pole of p ( z ) / q ( z ) q ( z 0 ) = 0 q ′ ( z 0 ) = 0 Res z 0 q p = q ′ ( z 0 ) p ( z 0 ) .
Worked example Residue checkFor z 2 + 1 z z = i p = z , q = z 2 + 1 , q ′ = 2 z Res i = 2 i i = 2 1
2 π i z 1 z 1 ⋅ d z = e i t 1 ⋅ i e i t d t = i d t t : 0 → 2 π i ⋅ 2 π = 2 π i 2 π i Every residue answer is a multiple of this one loop.
2 π i the value of one counterclockwise loop around a single simple spike of strength 1
Complex number z = x + iy
Euler spinner e to the it
Velocity z prime and step dz
Residue leftover strength
Residue theorem 2 pi i sum Res
Each box must be solid before the next: you cannot understand d z z ( t ) z ( t ) e i t C
What does z = x + i y A point (address) in the 2D complex plane; x y
What does ∣ z ∣ Distance from the origin;
∣ z ∣ = x 2 + y 2 .
Where does e i t t ∈ [ 0 , 2 π ] On the unit circle; starts and ends at 1
What is z ′ ( t ) z ( t ) = e i t i z ′ ( t ) = i e i t i 90°
Why is d z = z ′ ( t ) d t d z
Split f = u + i v u v Two ordinary real functions of ( x , y )
What does "analytic" mean in one sentence? The complex derivative exists and gives the same value from every direction (Cauchy–Riemann holds).
Why is z ˉ Its derivative depends on the direction of the step, so Cauchy–Riemann fails.
What is a residue? The coefficient of the 1/ ( z − z 0 )
Residue at a simple pole of p / q p ( z 0 ) / q ′ ( z 0 )
What is one counterclockwise loop around a single unit-strength spike worth? 2 π i
