Intuition The ONE core ideaA complex function is a rule that moves points around a flat sheet, and the "nice" ones (analytic) are the maps that, up close, only ever spin and zoom a tiny patch — never stretch it lopsided or flip it in a mirror. Every symbol below exists to make that single sentence precise and testable.
This page assumes you have seen nothing . We name and picture every ingredient the parent note leans on, in build-order, so that when you meet f ′ ( z 0 ) u x = v y
Definition A point in 2D — the pair( x , y ) origin O x y ( x , y )
x right (can be negative = left).y up (can be negative = down).
The picture: an arrow from O
We need this because a complex function eats a point of the sheet and spits out another point of the sheet. No sheet, no story.
i point of the plane as a single number we can add and multiply. Real numbers only fill a line. To get the second dimension we invent one new number i
Definition The imaginary uniti i i 2 = − 1 action : multiplying by i 1 i − 1 − i
Why we need it: it lets a 2D point be written as one algebraic object we can differentiate — that single-number-ness is the entire reason analytic functions are special.
z = x + i y ( x , y ) x plus y i
x real part , written Re ( z ) y imaginary part , written Im ( z ) y i C
Im ( z ) i Why it feels right: it's the "imaginary" part, so surely it carries i The flaw: Im ( x + i y ) = y i stripped off . Fix: real and imaginary parts are BOTH ordinary real numbers; i
For polar form, Euler's formula, and multiplication-as-rotation, see Complex numbers — polar form and Euler's formula — the parent uses e z = e x ( cos y + i sin y )
z ˉ = x − i y y → − y x
∣ z ∣ = x 2 + y 2 O z length of the arrow. By Pythagoras on the right triangle with legs x y
Why the topic needs both: the parent's Example 2 uses z ˉ mirror — orientation-reversing, so not nice), and Example 4 uses ∣ z ∣ 2 = x 2 + y 2 forbids .
Intuition Splitting one map into two heightmapsA complex function moves each input point ( x , y ) u v where you started , both u v x y
f ( z ) = u ( x , y ) + i v ( x , y )
f → u ( x , y ) v ( x , y ) u v ( x , y ) i
Why: the Cauchy–Riemann equations are exactly a coupling between these two terrains . You cannot state them without first having u v
u two inputs. To measure a slope we must fix one and wiggle the other — otherwise "the" slope is ambiguous. A partial derivative freezes all-but-one variable.
Definition Partial derivativeu x = ∂ x ∂ u u y x Δ x u u x = lim Δ x → 0 Δ x u ( x + Δ x , y ) − u ( x , y ) . slope of the slice you get by cutting the terrain with a plane held parallel to the x u y y
Why: the CR equations u x = v y u y = − v x between these four slopes . They are the topic's punchline; these are its letters.
For where these slopes assemble into a full local-linearisation matrix, see Jacobian and the multivariable chain rule .
"lim Δ z → 0 ( stuff ) Δ z stuff homes in on one fixed value. That value is the limit.
Intuition The twist that makes complex analysis hardand beautiful
On a line you can only shrink toward 0 left or right — two directions. On the plane Δ z = Δ x + i Δ y infinitely many directions: along the x y same number no matter the approach direction.
Definition The derivativef ′ ( z 0 ) f ′ ( z 0 ) = lim Δ z → 0 Δ z f ( z 0 + Δ z ) − f ( z 0 ) , required to give the same value for every approach direction. That single demand is what the parent note squeezes down the two axes to force the Cauchy–Riemann equations out.
Why direction-independence? Only then does "divide by Δ z Δ z f ′
Definition Matching partsTwo complex numbers are equal only when their real parts agree and their imaginary parts agree — they are two separate equations hiding in one. Picture: two points coincide only if they have the same right-coordinate AND the same up-coordinate.
Why: Step 3 of the parent sets u x + i v x = v y − i u y splits it into the two CR equations. Without "matching parts" that split is a mystery.
Imaginary unit i quarter turn
Complex number z = x + iy
Function f = u + i v two terrains
Partial derivatives u_x u_y v_x v_y
Limit and direction independence
Complex derivative f prime of z
Analytic functions the topic
Self-test: cover the right side and answer each before revealing.
What does the symbol i do to a point, geometrically? Rotates it 90° counter-clockwise about the origin (i 2 = − 1
In z = x + i y y An ordinary real number — the i Im ( z ) = y i y
What does z ˉ Mirrors it across the real axis (y → − y
What is ∣ z ∣ The distance from the origin to
z , i.e. the length of the arrow,
x 2 + y 2 .
Why do we split f u v The output is a 2D point; u v ( x , y )
What does the subscript in u x Freeze y x u
Why is a complex limit stricter than a real one? The wiggle Δ z 0
What rule lets one complex equation become two real equations? Two complex numbers are equal iff their real parts match AND their imaginary parts match.
