Exponential form z = re^(iθ) — Hinglish notes

3.5.7 · Maths › Complex Numbers

YE HAI KYA?

Iska core hai Euler's formula: $e^{i θ} = cos θ + i sin θ$

$e^{i θ} = cos θ + i sin θ$ AATA KAHAN SE HAI? (Scratch se derivation)

Formula Taylor series se derivation

Yahan se kyun shuru karein? Kyunki $e^{x}$ , $sin$ , $cos$ teeno ki known power series hain, aur hum bas $i θ$ ko $e^{x}$ mein feed karte hain aur pattern khud bann jaata hai.

Real exponential series se shuru karein (sabhi inputs ke liye valid): $e^{u} = 1 + u + \frac{u ^{2}}{2 !} + \frac{u ^{3}}{3 !} + \frac{u ^{4}}{4 !} + \dots$

Ye step kyun? Hum $e^{i θ}$ ko define karte hain usi series mein $u = i θ$ daalke — yahi complex inputs tak natural extension hai. $e^{i θ} = 1 + i θ + \frac{( i θ ) ^{2}}{2 !} + \frac{( i θ ) ^{3}}{3 !} + \frac{( i θ ) ^{4}}{4 !} + \dots$

$i$ ki powers use karein: $i^{2} = - 1, i^{3} = - i, i^{4} = 1$ (4 ka cycle). $= 1 + i θ - \frac{θ ^{2}}{2 !} - i \frac{θ ^{3}}{3 !} + \frac{θ ^{4}}{4 !} + \dots$

Ye step kyun? $i$ ke bina waale terms (real) aur $i$ ke saath waale terms (imaginary) alag karo: $= = c o s θ (1 - \frac{θ ^{2}}{2 !} + \frac{θ ^{4}}{4 !} - \dots) + i = s i n θ (θ - \frac{θ ^{3}}{3 !} + \frac{θ ^{5}}{5 !} - \dots)$

Dono brackets bilkul $cos θ$ aur $sin θ$ ki Taylor series hain. Isliye $e^{i θ} = cos θ + i sin θ$ aur $r$ se multiply karne par: $z = r (cos θ + i sin θ) = r e^{i θ}$ .

Multiplication = angles add karna KYU HOTA HAI

Worked examples

Common mistakes (Steel-manned)

Active recall

Recall Khud se test karo (answers chhupa lo)

$r$ kya represent karta hai? Answer: modulus $∣ z ∣ = x^{2} + y^{2}$ .
$θ$ kya represent karta hai? Answer: argument, +real axis se angle.
Euler's formula? $e^{i θ} = cos θ + i sin θ$ .
$∣ e^{i θ} ∣ =$ ? $1$ .
Multiplication mein angles kyun add hote hain? Kyunki $e^{a} e^{b} = e^{a + b}$ .

Recall Feynman: ek 12-saal ke bacche ko samjhao

Ek ghadi ki sui imagine karo jiska length $r$ hai aur jo "3 baje" waali jagah se shuru hoti hai. $θ$ number batata hai ki sui ko anticlockwise kitna ghumaana hai. Sui ki naok par tumhara complex number land karta hai. Jab tum aisi do numbers ko multiply karte ho, toh suiyan stretch hoti hain (unki lengths multiply) aur unke ghoomne add hote hain. Bus itna hi — ek complex number sirf "ek stretched, rotated arrow" hai, aur arrows multiply karne ka matlab hai stretch-and-turn.

Connections

Modulus and argument of a complex number
Polar form (trigonometric) r(cosθ + i sinθ)
De Moivre's theorem
Roots of unity
Euler's identity e^(iπ)+1=0
Taylor and Maclaurin series
Complex multiplication as rotation and scaling

Ek complex number ki exponential form kya hoti hai?

z = r e^{i θ}

jahan

r = ∣ z ∣

aur

θ = ar g z

Euler's formula batao.

e^{i θ} = cos θ + i sin θ

∣ e^{i θ} ∣

kya hai aur kyun?

1

, kyunki

cos^{2} θ + sin^{2} θ = 1

; ye unit circle par hota hai.

r_{1} e^{i θ_{1}} \cdot r_{2} e^{i θ_{2}}

multiply karne par kya hota hai?

Moduli multiply hote hain, arguments add hote hain:

r_{1} r_{2} e^{i (θ_{1} + θ_{2})}

Exponential form mein

z^{n}

likho (De Moivre).

z^{n} = r^{n} e^{in θ}

1 + i

ko exponential form mein convert karo.

2 e^{iπ /4}

- 1 + i 3

ko exponential form mein convert karo.

2 e^{i 2 π /3}

(second quadrant!).

Argument ke liye

arctan (y / x)

blindly kyun nahi use kar sakte?

Ye sirf

(- π /2, π /2)

output deta hai;

x, y

ke signs se quadrant fix karna padta hai.

Series se

e^{i θ} = cos θ + i sin θ

kaise derive hota hai?

u = i θ

e^{u}

ki Taylor series mein daalo; real terms

cos θ

banate hain, imaginary terms

sin θ

banate hain.

Exponential form se

(1 + i)^{8}

kya hai?

(2)^{8} e^{i 2 π} = 16