Cauchy's integral theorem and formula — Hinglish notes

4.10.3 · Maths › Advanced Topics (Elite Level)

1. Setup: hum kaunse objects ki baat kar rahe hain?

YEH kyun important hai: real differentiability sirf left/right check karti hai. Complex differentiability demand karti hai ki same limit har direction se aaye — yeh kaafi zyada strong constraint hai. Yahi extra rigidity neeche diye gaye har theorem ki engine hai.

2. Cauchy–Riemann equations (yahan se rigidity janam leti hai)

$f (z) = u (x, y) + i v (x, y)$ likho jahan $z = x + i y$ .

Limit ko real axis ke along approach karo ( $h = Δ x$ real): $f^{'} (z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}$

Imaginary axis ke along approach karo ( $h = i Δ y$ ): $f^{'} (z) = \frac{1}{i} (\frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y}) = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y}$

Yeh dono equal hone chahiye, toh real aur imaginary parts match karo:

YEH key lemma kyun hai: yeh equations exactly woh condition hain jo contour integral ko curl-free banati hain, aur issi wajah se loop integral zero ho jaata hai.

3. Cauchy's Theorem scratch se derive karna

ISKO hum kaise derive karte hain (Green's theorem ke zariye — honest route):

$f = u + i v$ , $d z = d x + i d y$ likho. Tab $\oint_{γ} f d z = \oint_{γ} (u + i v) (d x + i d y) = real part \oint (u d x - v d y) + i imag part \oint (v d x + u d y)$

Dono par Green's theorem $\oint (P d x + Q d y) = \iint_{R} (\partial_{x} Q - \partial_{y} P) d A$ apply karo:

Real part: $P = u, Q = - v$ deta hai $\iint (- v_{x} - u_{y}) d A$ .
Imag part: $P = v, Q = u$ deta hai $\iint (u_{x} - v_{y}) d A$ .

"Simply connected" non-negotiable kyun hai: Green's theorem ko chahiye ki $f$ poori region $R$ par holomorphic ho jo $γ$ se enclosed ho. Ek bhi hole (ek point jahan $f$ blow up ho) aur argument toot jaata hai — aur exactly issi wajah se agla formula nonzero hai.

4. Cauchy's Integral Formula

ISKO hum kaise derive karte hain: Integrand $\frac{f ( z )}{z - a}$ $γ$ ke andar har jagah holomorphic hai siway $z = a$ ke. $γ$ ko ek chhote circle $C_{ε}$ mein deform karo radius $ε$ ka jo $a$ par centered ho (yeh legal hai, kyunki dono ke beech ke region mein integrand holomorphic hai → Cauchy's theorem se difference vanish ho jaata hai): $\oint_{γ} \frac{f ( z )}{z - a} d z = \oint_{C_{ε}} \frac{f ( z )}{z - a} d z .$

$C_{ε}$ par $z = a + ε e^{i θ}$ , $d z = i ε e^{i θ} d θ$ rakho: $\oint_{C_{ε}} \frac{f ( z )}{z - a} d z = \int_{0}^{2 π} \frac{f ( a + ε e ^{i θ} )}{ε e ^{i θ}} i ε e^{i θ} d θ = i \int_{0}^{2 π} f (a + ε e^{i θ}) d θ .$

$ε \to 0$ lene do; continuity se $f (a + ε e^{i θ}) \to f (a)$ : $= i \cdot 2 π f (a) = 2 π i f (a) .$

$2 π i$ se divide karo. Ho gaya.

Figure — Cauchy's integral theorem and formula

5. Worked Examples

6. Common Mistakes (Steel-manned)

7. Flashcards

Simply connected domain par holomorphic

f

ke liye

\oint_{γ} f d z

ki value kya hai?

0

(Cauchy's Integral Theorem).

Cauchy's Integral Formula state karo.

f (a) = \frac{1}{2 π i} \oint_{γ} \frac{f ( z )}{z - a} d z

jahan

a

γ

ke andar ho.

Cauchy's theorem ke liye kaun si do conditions chahiye?

f

region par holomorphic ho, aur domain simply connected ho (koi enclosed singularity nahi).

Kaun sa theorem

\oint f d z

ko ek double integral mein reduce karta hai jo Cauchy–Riemann se vanish ho jaata hai?

Green's theorem.

Cauchy–Riemann equations state karo.

u_{x} = v_{y}

aur

u_{y} = - v_{x}

\oint_{∣ z ∣ = 1} \frac{d z}{z}

ki value?

2 π i

n

-th derivative ke liye generalized formula?

f^{(n)} (a) = \frac{n !}{2 π i} \oint_{γ} \frac{f ( z )}{( z - a ) ^{n + 1}} d z

Holomorphic functions infinitely differentiable kyun hoti hain?

Integral formula

f^{(n)}

f

ke contour integral se express karta hai, jo sab

n

ke liye valid hai.

\oint_{∣ z ∣ = 2} \frac{e ^{z}}{z - 1} d z

compute karo.

2 π i e

Recall Feynman: 12-saal ke bachche ko samjhao

Ek perfectly smooth trampoline imagine karo jisme koi rips nahi hain. Agar tum uspe ek closed loop mein chalo aur jo tilts feel karo unhe add karte jao, toh tum exactly wahin pahunch jaate ho jahan se shuru kiya tha — total change zero hai. Yahi Cauchy's theorem hai. Ab creepy part: agar trampoline ek fenced circle ke andar smooth hai, toh sirf fence ke saath height feel karke tum andar kisi bhi point ki exact height calculate kar sakte ho — andar kadam rakhne ki zarurat hi nahi! Yahi integral formula hai. Smooth complex functions itni well-behaved hoti hain ki edge secretly poora inside jaanti hai.

Connections

Cauchy-Riemann equations — woh lemma jo sab kuch power karta hai.
Green's theorem — contour se area integral tak ka bridge.
Residue theorem — kaafi poles ke liye grand generalization.
Laurent series — yahan se $1/ (z - a)^{n + 1}$ terms aate hain.
Liouville's theorem — bounded entire functions constant hoti hain, iska direct corollary.
Maximum modulus principle — rigidity ka ek aur consequence.