Complex analysis — analytic functions, Cauchy-Riemann equations

WHAT we are studying

WHAT: A function $f:\mathbb{C}\to\mathbb{C}$ , written $f(z)=u(x,y)+i\,v(x,y)$ where $z=x+iy$ .
WHY it matters: Analytic functions are the "smooth, structure-preserving" maps of the plane. They preserve angles (conformal), satisfy Laplace's equation, and are infinitely differentiable + locally given by a power series. This is the gateway to contour integration, residues, and physics (fluid flow, electrostatics, signal processing).
HOW we test analyticity: Check the Cauchy–Riemann (CR) equations + continuity of partials.

Deriving the Cauchy–Riemann equations from scratch

Write $f(z)=u(x,y)+i\,v(x,y)$ and $\Delta z=\Delta x+i\,\Delta y$ .

Step 1 — Approach along the real axis ( $\Delta y=0$ , $\Delta z=\Delta x$ ). Why this step? It isolates how $f$ changes when only $x$ moves. $f'(z)=\lim_{\Delta x\to0}\frac{u(x+\Delta x,y)+iv(x+\Delta x,y)-u-iv}{\Delta x}=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}.$

Step 2 — Approach along the imaginary axis ( $\Delta x=0$ , $\Delta z=i\,\Delta y$ ). Why this step? Same derivative, but we slide in the orthogonal direction. Note $1/(i\,\Delta y)=-i/\Delta y$ . $f'(z)=\lim_{\Delta y\to0}\frac{u(x,y+\Delta y)+iv(x,y+\Delta y)-u-iv}{i\,\Delta y}=\frac{1}{i}\left(\frac{\partial u}{\partial y}+i\frac{\partial v}{\partial y}\right)=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}.$

Step 3 — Demand the two results are equal. Why this step? That is literally what "derivative independent of direction" means. $\underbrace{\frac{\partial u}{\partial x}}_{\text{real}}+i\underbrace{\frac{\partial v}{\partial x}}_{\text{imag}}=\underbrace{\frac{\partial v}{\partial y}}_{\text{real}}-i\underbrace{\frac{\partial u}{\partial y}}_{\text{imag}}.$

Match real parts and imaginary parts separately:

Figure — Complex analysis — analytic functions, Cauchy-Riemann equations

The sufficiency direction (the warning everyone forgets)

CR equations alone are necessary but not sufficient. The clean theorem:

Consequence: harmonic functions

Worked examples

Steel-manning common mistakes

Recall Feynman: explain to a 12-year-old

Imagine a stretchy rubber sheet with arrows drawn on it (a map). A "nice" map can only spin and zoom little neighborhoods — it can twist and grow them, but it must keep all the angles the same and never flip them inside-out (no mirror). The Cauchy–Riemann equations are just the secret handshake that says "yes, this map only spins-and-zooms." If a map flips things over (like a mirror), or stretches more sideways than up-down, it breaks the handshake and isn't "nice."

Active recall

What does it mean for

f

to be complex-differentiable at

z_0

The limit

\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}

exists and is the same for every direction of

\Delta z\to0

State the Cauchy–Riemann equations.

u_x=v_y

and

u_y=-v_x

, where

f=u+iv

Why do the CR equations arise?

Equating

f'

computed along the real axis (

u_x+iv_x

) with

f'

along the imaginary axis (

v_y-iu_y

Are CR equations sufficient for analyticity?

No — only necessary. You also need the partials

u,v

to be continuous on a neighborhood.

Give a function that satisfies CR at exactly one point but is nowhere analytic.

f(z)=|z|^2=x^2+y^2

(CR holds only at

z=0

, not on an open set).

Why is

f(z)=\bar z

not analytic?

u=x,v=-y

give

u_x=1\ne v_y=-1

; CR fails everywhere (it's a reflection, orientation-reversing).

What PDE do the real and imaginary parts of an analytic function satisfy?

Laplace's equation

u_{xx}+u_{yy}=0

(and same for

v

); they are harmonic conjugates.

Two expressions for

f'(z)

in terms of partials?

f'=u_x+iv_x=v_y-iu_y

What is the Jacobian form forced by analyticity?

\begin{pmatrix}a&-b\\ b&a\end{pmatrix}

, i.e. rotation × scaling = multiplication by

a+ib

What is

f'

for

f(z)=e^z

and why?

e^z

; from

u=e^x\cos y,v=e^x\sin y

f'=u_x+iv_x=e^x\cos y+ie^x\sin y=e^z

Connections

Complex numbers — polar form and Euler's formula
Conformal mappings
Harmonic functions and Laplace's equation
Contour integration and Cauchy's integral theorem
Power series and radius of convergence
Jacobian and the multivariable chain rule

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, complex function $f(z)=u(x,y)+iv(x,y)$ basically ek 2D point ko doosre 2D point pe map karta hai. Magic tab hota hai jab uska derivative ek single complex number ke roop mein exist kare — yaani aap kisi bhi direction se $\Delta z\to0$ le jao, answer same aaye. Yeh "direction se farak nahi padta" wali demand bahut strong hai, aur isi se nikalti hain Cauchy–Riemann equations: $u_x=v_y$ aur $u_y=-v_x$ .

Derivation simple hai: real axis ke along derivative nikaalo to milta hai $u_x+iv_x$ ; imaginary axis ke along nikaalo to $1/i=-i$ ke kaaran milta hai $v_y-iu_y$ . Dono equal hone chahiye, to real part = real part aur imaginary part = imaginary part — bas wahi se CR equations aa gaye. Yaad rakho: ek minus sign zaroor aata hai (cross-pair clash), warna sab galat ho jayega.

Ek important trap: sirf CR satisfy hona kaafi nahi hai. Aapko partial derivatives ka continuous hona bhi chahiye, aur CR ek pure region (open set) pe hold karna chahiye. Jaise $f(z)=|z|^2$ sirf $z=0$ pe CR satisfy karta hai — isliye woh kahin bhi analytic nahi, kyunki ek single point open set nahi hota. Aur $f(z)=\bar z$ (reflection) to kahin bhi analytic nahi, kyunki woh orientation flip karta hai.

Geometry mein samjho: analytic map locally sirf ghuma (rotate) aur zoom (scale) kar sakta hai — angles preserve karta hai (conformal), mirror-flip allowed nahi. Isi wajah se uska Jacobian $\begin{pmatrix}a&-b\\b&a\end{pmatrix}$ banta hai, jo $a+ib$ se multiply karne ke barabar hai. Bonus: $u$ aur $v$ dono Laplace equation satisfy karte hain — yeh physics (fluid flow, electrostatics) se direct connect karta hai.

Complex analysis — analytic functions, Cauchy-Riemann equations

WHAT we are studying

Deriving the Cauchy–Riemann equations from scratch

The sufficiency direction (the warning everyone forgets)

Consequence: harmonic functions

Worked examples

Steel-manning common mistakes

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Advanced Topics (Elite Level)

Connections