Cauchy's integral theorem and formula: Foundations

4.10.3 · D1 · Maths › Advanced Topics (Elite Level) › Cauchy's integral theorem and formula

Parent note padhne se pehle, aapko dekhna hoga ki har symbol ka matlab kya hai. Hum unhe order mein build karte hain, taaki har naya symbol sirf pehle se bane huon ko use kare.

1. $C$ — the complex plane (woh stage jahan sab kuch rehta hai)

Plain words mein: $z$ ko ek flat map par ek address ki tarah socho. Aap $x$ steps east aur $y$ steps north chalte hain. Woh landing spot hi woh number hai. Set $C$ simply "map par saare possible addresses" hai.

Picture: har complex number ek plane mein ek dot hai. Horizontal axis "real part" $x$ ko rakhta hai (likha $Re z$ ), vertical axis "imaginary part" $y$ ko rakhta hai (likha $Im z$ ).

Figure — Cauchy's integral theorem and formula

Figure s01 — the complex plane: ek amber arrow origin se dot $z = x + i y$ tak jaata hai; ek dashed cyan line horizontal axis tak drop hoti hai jo $x = Re z$ mark karti hai, doosri vertical axis tak left jaati hai jo $y = Im z$ mark karti hai.

Topic ko yeh kyun chahiye: Cauchy's theorem is plane mein ek path par integrate karta hai. Agar $z$ sirf ek line par ek number hota, toh koi loops nahi hote jiske around walk kiya ja sake aur prove karne ke liye kuch nahi hota. Yeh poora subject exist karta hai kyunki $C$ two-dimensional hai.

2. $e$ , the exponential, aur $e^{i θ}$ — use karne se pehle

$∣ z ∣$ aur $θ$ ko hum theek se agli section mein define karte hain; yahaan hum sirf symbol $e^{i θ}$ earn karte hain taaki baad mein use ho sake.

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

Picture: origin $0$ se dot $z$ tak arrow draw karo. Uski length $∣ z ∣$ hai; uski tilt $θ$ hai. Yeh wahi right-triangle idea hai jo real vectors ke liye use hoti hai — $x$ adjacent side hai, $y$ opposite side.

Figure s02 — cyan mein unit circle; length $1$ ka ek amber arrow angle $θ$ par point $e^{i θ} = cos θ + i sin θ$ par land karta hai. Origin par ek small white arc angle $θ$ mark karta hai.

Topic ko yeh kyun chahiye: integral formula ek tiny circle ko $z = a + ε e^{i θ}$ parametrise karta hai. Woh notation pure polar hai: $ε$ modulus (radius) hai aur $θ$ argument ko poora sweep karta hai. Polar form ke bina aap key step bhi nahi likh sakte.

4. Ek function $f : C \to C$ — ek rule jo points move karta hai

Plain words mein: ek dot daalo, doosri dot milo. Puri plane ko uthaya jaata hai, stretch, twist, aur wapas rakha jaata hai.

Topic ko yeh kyun chahiye: parent $f = u + i v$ aur $d z = d x + i d y$ likhta hai, phir loop integral ko ek real part aur ek imaginary part mein split karta hai. Woh split yahi $u, v$ decomposition hai $d z = d x + i d y$ ke saath combine hoke. Green's-theorem step ko $d z = d x + i d y$ ke bina follow nahi kar sakte dono ke bina.

5. Derivative as a limit — aur direction kyun matter karta hai

Picture: point $z_{0}$ par khade ho. Ek tiny step $h$ le ke nearby point par jao. Poochho: output kitna move hua, per unit of input step? Woh ratio $\frac{f ( z _{0} + h ) - f ( z _{0} )}{h}$ local stretch-and-turn factor hai.

Figure s03 — point $z_{0}$ (amber dot) jisme aath cyan arrows $h$ saari directions mein fan out kar rahe hain; figure ka caption stress karta hai ki limit unme se har ek along same value deni chahiye.

Topic ko yeh kyun chahiye — aur yeh tool kyun: Real calculus mein derivative sirf left aur right check karta hai (do directions). Yahaan $h$ ek plane mein rehta hai, isliye infinitely many approach directions hain. Sabke liye ek jawab demand karna ek colossal constraint hai. Wahi constraint Cauchy–Riemann equations ko force karti hai, jo loop integrals ko vanish karati hain. Hum limit use karte hain kyunki "instantaneous stretch factor" ka koi meaning nahi hai bina step ko zero tak shrink kiye.

Intuition Woh do directions jo Cauchy–Riemann create karte hain

Kyunki har direction same $f^{'} (z_{0})$ deni chahiye, particularly horizontal approach aur vertical approach agree karni chahiye. Un do chosen directions ko dekhna hi woh equations pin karta hai jo agli section mein hain.

Horizontal approach — let $h = Δ x$ ek tiny real step ho (pure east). Sirf $x$ change hota hai, isliye $f^{'} (z_{0}) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} .$ Vertical approach — let $h = i Δ y$ ek tiny imaginary step ho (pure north). Ab $h$ mein aage $\frac{1}{i} = - i$ factor hai, jo deta hai: $f^{'} (z_{0}) = \frac{1}{i} (\frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y}) = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y} .$ Inhe equal set karna — real part to real part, imaginary part to imaginary part — exactly wahi hai jo §6 mein Cauchy–Riemann equations force karta hai. (Symbols $\partial u / \partial x$ etc. bilkul neeche define hain; agar woh naye hain toh pehle §6 padho, phir wapas aao.)

6. Partial derivatives $\partial u / \partial x$ — ek chosen direction mein slope

Picture: $u (x, y)$ plane par ek height landscape hai. Due east chalo — ground kitni steeply utha? Woh hai $\partial u / \partial x$ . Due north chalo — woh hai $\partial u / \partial y$ .

Topic ko yeh kyun chahiye: §5 mein horizontal approach $\partial_{x}$ use karta hai; vertical approach $\partial_{y}$ use karta hai. Do directional limits ko term-by-term match karna the Cauchy–Riemann equations $u_{x} = v_{y}, u_{y} = - v_{x}$ produce karta hai (pehla real parts pair karta hai, doosra imaginary parts). Partial derivatives woh language hai jin mein woh equations likhi hain.

7. $γ$ , the contour — ek path jo aap walk karte hain

Picture: complex plane par ek bent, possibly looping arrow-path draw ki. Chhota arrowhead travel direction dikhata hai.

Topic ko yeh kyun chahiye: $\oint_{γ}$ matlab hai "loop $γ$ ke along add karo." Integral sign par circle symbol, $\oint$ , specifically matlab hai path khud par close hoti hai. Cauchy jo bhi prove karta hai woh inhi loops ke baare mein hai.

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

Picture: path par har point par, ek arrow $f (z)$ aapko ek local push batata hai; $d z$ woh direction hai jis mein aap step karte hain. Unka product ek contribution hai; integral loop ke around saare contributions ka grand total hai.

Figure s04 — ek cyan closed loop $γ$ ek direction arrowhead ke saath (counter-clockwise); loop ke tangent chhe amber arrows tiny steps $d z$ mark karte hain, har ek sample point par, illustrating "loop ke around har $d z$ ke saath $f (z)$ times sum karo."

Topic ko yeh kyun chahiye: yahi woh object hai jise Cauchy's theorem zero set karta hai. Ise "tiny steps ka weighted sum" ki tarah samajhna hi "total cancel hokar $0$ ho jaata hai" ko believable banata hai, naki magic.

9. Singularities, regions, aur "simply connected"

Picture: ek solid pancake (theek hai) versus ek pancake jisme ek bite missing hai (bite ko hug karne wala loop phansa hua hai).

Topic ko yeh kyun chahiye: Green's-theorem step demand karta hai ki $f$ $γ$ ke poore filled interior par holomorphic ho. Andar ek bhi singularity ek "hole" hai jo simple connectivity tod deti hai, aur proof wahaan collapse ho jaata hai. Yahi exactly reason hai ki $\oint d z / z$ not zero hai — interior mein origin par singularity hai.

Prerequisite map (ise kaise padhein)

Har box ek foundation hai. Har arrow ko sentence padho "tail par box ka sense hona zaroori hai head par box ka sense hone se pehle." Arrows upar ke boxes se neeche follow karo; sab kuch eventually bottom box mein feed hota hai — parent topic.

Har arrow kehta hai "left box ka sense hona zaroori hai right box ka sense hone se pehle." Teeno streams — $C$ ki algebra, $f$ ki calculus, aur paths ki geometry — sab parent topic Cauchy's integral theorem and formula par merge hote hain.

Yeh aage kyun feed karte hain (ek line each)

Cauchy-Riemann equations §5 (do directional limits) + §6 (partials) se paida hote hain.
Green's theorem woh hai jo §8 ke loop integral ko ek area integral mein convert karta hai jise CR equations khatam kar deti hain.
Jab loops vanish ho jaate hain, contour ko ek tiny circle mein deform karna integral formula deta hai, aur $(z - a)$ ki higher powers ke saath trick repeat karna Residue theorem deta hai.
§5 mein jo rigidity aap dekhte hain woh baad mein Liouville's theorem aur Maximum modulus principle ko power karti hai.

Equipment checklist

Right side cover karo aur khud test karo.

Symbol

C

kaunsa set denote karta hai?

C = {x + i y ∣ x, y \in R}

— saare complex numbers ka field.

i^{2}

kya hai, aur

i

se multiply karna geometrically kya karta hai?

i^{2} = - 1

;

i

se multiply karna ek point ko 90° counter-clockwise rotate karta hai.

z = x + i y

ke liye

Re z

aur

Im z

kya hain?

Re z = x

(horizontal part),

Im z = y

(vertical part).

e

kya hai aur exponential

e^{x}

kya define karta hai?

e \approx 2.71828

;

e^{x}

woh function hai jo apni slope ke barabar hai, series

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

se build hota hai.

Euler's form state karo aur

e^{i θ}

geometrically kya hai?

e^{i θ} = cos θ + i sin θ

; yeh angle

θ

par unit circle par woh point hai.

Argument

θ

multi-valued kyun hai, aur hum ise kaise pin karte hain?

Koi bhi poora turn

2 π

add karna same point par land karta hai; hum principal value

- π < θ \leq π

fix karte hain ek branch cut use karke.

d z

d x

aur

d y

ke terms mein likho.

d z = d x + i d y

— ek tiny east step plus

i

times ek tiny north step.

Agar

f (z) = u + i v

, toh

u

aur

v

kya hain?

x, y

ke real functions:

u

output ka real part hai,

v

imaginary part.

Complex derivative real se stricter kyun hai?

Limit ko plane mein har direction se same value deni chahiye, sirf left/right nahi.

Cauchy–Riemann equations kaunse do approach directions produce karte hain?

Horizontal (

h = Δ x

) aur vertical (

h = i Δ y

) approaches, equal set kiye hue.

\partial u / \partial x

kya measure karta hai?

u

ki slope jab sirf

x

nudge kiya jaaye aur

y

fixed rakha jaaye.

\oint_{γ}

par circle kya signify karta hai?

Contour

γ

closed hai — woh wahaan khatam hota hai jahaan shuru hua (ek loop).

Words mein,

\oint_{γ} f d z

kya compute karta hai?

Loop ke around poore raaste

f (z)

times har tiny path-step

d z

ka sum.

Kya cheez ek point ko

f

ki singularity banati hai?

f

wahaan holomorphic fail hota hai — typically

f (z)

blow up karta hai, jaise

1/ z

z = 0

Complex analysis mein ek region (domain) kya hai?

Ek set jo open ho (koi boundary points nahi) aur connected ho (koi bhi do points interior path se joinable).

"Simply connected" ka matlab kya hai?

Region mein har loop ko ek point tak shrink kiya ja sake bina ise chhoode — koi holes nahi.

Ek enclosed singularity Cauchy's theorem kyun tod deti hai?

Woh ek hole punch karti hai, isliye interior aur simply connected nahi rehta aur Green's theorem poore region par apply nahi ho sakta.