Laurent series — principal part, annulus of convergence: Foundations

4.10.4 · D1 · Maths › Advanced Topics (Elite Level) › Laurent series — principal part, annulus of convergence

Parent note ki ek bhi line padhne se pehle, tumhe lagbhag ek dozen chote vocabulary pieces khud ke hone chahiye. Yeh page unme se har ek ko zero se banata hai, ek aisi order mein jahan har cheez pichli par tikti hai. Neeche kuch bhi assume nahi kiya gaya — agar parent note ne koi symbol use kiya, toh woh pehle yahan define kiya gaya hai.

0. Complex plane — jahan sab kuch rehta hai

Figure — Laurent series — principal part, annulus of convergence

Figure s01 — Alt text: The complex plane with horizontal real axis and vertical imaginary axis. A lavender arrow runs from the origin to the point $z = x + i y$ . A butter-yellow segment along the bottom marks the horizontal distance $x = Re (z)$ ; a mint segment up the side marks the vertical distance $y = Im (z)$ ; a coral dashed line drops from the point down to the real axis. The length of the lavender arrow is labelled $∣ z ∣ = x^{2} + y^{2}$ .

$x$ real part hai, likha jaata hai $Re (z)$ — horizontal coordinate (woh butter-yellow segment).
$y$ imaginary part hai, likha jaata hai $Im (z)$ — vertical coordinate (woh mint segment).

Yeh kyun chahiye: is chapter ka har function plane ka ek point $z$ khaata hai aur doosra complex number return karta hai. Yahan koi number line nahi hai — yahan poora plane hai — aur woh extra dimension exactly wahi hai jiske karan singularities baad mein "rings" ke roop mein fence off ki ja sakti hain.

Yeh kyun chahiye: annulus ka poora idea hai "woh sab points jinki $a$ se distance $r$ aur $R$ ke beech hai." Woh distance ek modulus hai. Jab bhi $∣ z - a ∣$ dekho, usse zor se padho " $z$ se fixed point $a$ tak ki distance."

1. Complex exponential — angles ko points mein convert karna

Polar form (agla section) symbol $e^{i θ}$ par lean karti hai. Isse use karne se pehle, hume kehna hoga ki iska matlab kya hai, kyunki $e$ ko ek imaginary power tak raise karna koi cheez nahi hai jo ordinary arithmetic explain kare.

Yeh kyun chahiye: is chapter ka har loop ek circle hai, aur $e^{i θ}$ ek point ko circle ke around drive karne ka sabse clean tarika hai by turning one dial $θ$ . Yahi wajah hai ki parent ke contour proof mein $e^{i θ}$ ke saath likha gaya hai.

2. Polar form — circles mein jaane ka secret weapon

Figure s02 — Alt text: The complex plane with a mint dashed unit circle centred at the origin. A coral arrow of length one points at angle $θ$ (this is $e^{i θ} = cos θ + i sin θ$ , landing on the unit circle). A longer lavender arrow along the same direction reaches the point $z = ρ e^{i θ}$ , its length labelled $ρ = ∣ z ∣$ . A butter-yellow arc near the origin marks the angle $θ$ swept up from the positive real axis.

Ab proof mein milne wala tiny movement rule — uski derivation ke saath:

Isse picture ki tarah padhna: $θ$ ko thoda nudge karna $z$ ko circle ke along slide karta hai; factor $i$ radial arrow $e^{i θ}$ ko $9 0^{\circ}$ se turn karta hai toh step circle ke tangent hota hai (exactly woh direction jisme tum around jaate waqt move karte ho).

3. Analytic — "nice, koi trouble nahi yahan" ke liye word

Yeh kyun chahiye: Cauchy integral formula aur Taylor's theorem sirf wahan kaam karte hain jahan $f$ analytic hai. Laurent series exactly wahan exist karti hai jahan analyticity fail hoti hai — toh tumhe pehle "nice" ka matlab pata hona chahiye taaki tum "not nice" ke baare mein baat kar sako.

4. Singularity, pole, essential — trouble-spots

Figure s03 — Alt text: A 3-D surface plot over the complex plane (horizontal axes $Re (z)$ and $Im (z)$ ) whose height is $∣ f (z) ∣$ for $f (z) = 1/ z$ . Away from the origin the surface is a gently sloping valley; directly above $z = 0$ it rockets upward into a tall spike labelled "blow-up", picturing the singularity.

Figure mein height $∣ f (z) ∣$ ko ek surface ki tarah show kiya gaya hai. Ek singularity par yeh spike ki tarah upar shoot karta hai.

Poori picture ke liye Poles and singularities dekho. Principal part ka poora point yeh hai ki woh singularity kitni buri hai usse measure karta hai negative-power terms count karke.

5. Series aur summation symbol $\sum$

Yeh kyun chahiye: Taylor par Laurent ki poori innovation hai: negative powers allow karo. Negative-power terms "principal part" hain. Game ka naam bas is split se zyada kuch nahi hai.

6. Geometric series — woh engine jo har expansion banata hai

7. Contour integral $\oint_{γ}$ , ek fact, aur residue

Figure s04 — Alt text: The complex plane showing a shaded lavender ring (annulus) around a fixed centre point $a$ marked with a slate dot. A coral inner circle has radius $r$ (a coral arrow from $a$ marks that inner radius); a lavender outer circle has radius $R$ . A mint dashed circle sits inside the ring as the loop $γ$ , with a mint arrowhead showing it is traversed anticlockwise (positive orientation).

Yeh kyun chahiye: yeh woh "sieve" hai jo parent ki derivation mein ek baar mein ek coefficient nikalta hai. Sirf power $k = - 1$ ek residue chhod jaata hai; baaki sab ek full loop par zero tak integrate ho jaate hain.

8. Annulus — woh ring jis par yeh series jeeti hain

Inner radius $r$ = $a$ se $f$ ki sabse kareeb singularity ki distance (woh jo beech mein hole fence off karti hai). Outer radius $R$ = $a$ se agali singularity ki distance jo aur door hai. Annulus of convergence dekho. Unke beech, $f$ analytic hai, aur woh ring exactly wahan hai jahan uski Laurent series converge karti hai.

Pieces topic ko kaise feed karte hain

Neeche ka diagram ek dependency map hai: ise top-to-bottom padho, jahan har arrow ka matlab hai "tail wali cheez tumhe pehle chahiye taaki head wali cheez kar sako." Agar tumhara renderer diagram nahi draw karta, toh yahi information words mein hai: complex plane ( $z = x + i y$ ) tumhe modulus (distance) deta hai, jo annulus banata hai. Complex exponential $e^{i θ}$ polar form deta hai, jo tumhe $a$ ke around loop karne deta hai; contour fact (sirf $k = - 1$ survive karta hai) ke saath milke yeh coefficient formula aur residue deta hai. Alag se, analytic define karta hai ki singularity kya hai, aur poles/essential ise classify karte hain. Geometric series woh algebra engine hai jo actual expansions produce karti hai. Charon strands Laurent series par converge karte hain.

Equipment checklist

Khud ko test karo — parent note padhne se pehle tum har ek ka jawab de pao.

Fixed symbol

a

is chapter mein kya denote karta hai?

Complex plane ka ek chosen, fixed point — woh centre jiske around hum expand karte hain aur jisse sabhi distances

∣ z - a ∣

measure karte hain.

∣ z - a ∣

ka ek plain phrase mein kya matlab hai?

Plane mein point

z

se fixed point

a

tak ki distance.

Euler's formula state karo aur

e^{i θ}

geometrically kaisa dikhta hai.

e^{i θ} = cos θ + i sin θ

; yeh unit circle par woh point hai jo angle

θ

anticlockwise chalke pahuncha jaata hai.

z

ko polar form mein likho aur batao har symbol kya hai.

z = ρ e^{i θ}

;

ρ = ∣ z ∣ \geq 0

0

se distance hai, aur

θ

positive real axis se angle hai (

2 π

ke multiples add karne tak defined).

Dikhao ki

\frac{d}{d θ} e^{i θ} = i e^{i θ}

cos θ + i sin θ

ko differentiate karo taaki

- sin θ + i cos θ = i (cos θ + i sin θ)

mile.

"Analytic" (holomorphic) ka precisely kya matlab hai?

Point ke around ek chote disk mein har jagah complex-differentiable; equivalently, wahan apni khud ki Taylor series ke equal.

Complex plane mein disk kya hai?

Centre ke

R

distance ke andar ke sab points ka filled set,

{z : ∣ z - a ∣ < R}

"Pole of order

m

" ki precise definition do.

Sabse chota

m \in N

jaise ki

(z - a)^{m} f (z)

a

par analytic aur nonzero ho.

Geometric series aur uski convergence condition state karo.

\frac{1}{1 - w} = \sum_{n \geq 0} w^{n}

, valid sirf tab jab

∣ w ∣ < 1

Ek circle parametrise karo aur

\oint_{γ} (z - a)^{k} d z

evaluate karo.

z = a + ρ e^{i θ}

ke saath yeh

i ρ^{k + 1} \int_{0}^{2 π} e^{i (k + 1) θ} d θ

ban jaata hai,

k = - 1

hone par

2 π i

aur warna

0

deta hai.

Full Laurent series aur uska coefficient formula likho.

f (z) = \sum_{n = - \infty}^{\infty} c_{n} (z - a)^{n}

jahan

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

Laurent coefficients ke terms mein residue

Res_{f} (a)

kya hai?

Yeh

c_{- 1}

hai,

\frac{1}{z - a}

term ka coefficient.

Annulus ki shape kya hai, kya uski boundaries included hain, aur uske do radii ka kya matlab hai?

Ek ring/donut

r < ∣ z - a ∣ < R

jisme dono rims excluded hain (strict inequalities);

r

a

se nearest singularity ki distance hai,

R

agali wali ki.

Laurent series ko Taylor series se kya alag karta hai?

Laurent series negative powers

(z - a)^{- n}

bhi allow karti hai.