4.10.1 · D1Complex analysis — analytic functions, Cauchy-Riemann equati
Ek complex function ek aisi rule hai jo points ko ek flat sheet par idhar-udhar move karti hai, aur jo "acchi" functions hain (analytic) — woh aisi maps hain jo, kareeb se dekho toh, ek chhoti si patch ko sirf spin aur zoom karti hain — use kabhi ek taraf se stretch nahi karti, na hi mirror mein flip karti hain. Neeche diye gaye har symbol ka kaam sirf yahi hai ki us ek sentence ko precise aur testable banaye.
Is page par yeh maana gaya hai ki tumne pehle kuch nahi dekha. Hum har ingredient ko naam denge aur picture banayenge jis par parent note rely karta hai, build-order mein — taaki jab tum f ′ ( z 0 ) u x = v y
Definition 2D mein ek point — pair( x , y ) origin O x y ( x , y )
x daayein (negative ho sakta hai = baayein).y upar (negative ho sakta hai = neeche).
Picture: O
Yeh isliye chahiye kyunki ek complex function sheet ka ek point khaati hai aur sheet ka doosra point bahar ugalti hai. Sheet nahi, toh kahani nahi.
i point ko ek single number ki tarah treat karein jise hum add aur multiply kar sakein. Real numbers sirf ek line fill karte hain. Doosra dimension paane ke liye hum ek naya number i
Definition Imaginary uniti i i 2 = − 1 action ki tarah padho: i 1 i − 1 − i
Isko kyun chahiye: yeh ek 2D point ko ek algebraic object ki tarah likhne deta hai jise hum differentiate kar sakein — yahi single-number-ness woh poori wajah hai jis se analytic functions special hain.
z = x + i y ( x , y ) x plus y i
x real part , likha jaata hai Re ( z ) y imaginary part , likha jaata hai Im ( z ) y i C
Im ( z ) i Yeh sahi kyun lagta hai: yeh "imaginary" part hai, toh zaroor i Galti: Im ( x + i y ) = y i strip off ho jaata hai. Fix: real aur imaginary parts DONO ordinary real numbers hain; i
Polar form, Euler's formula, aur multiplication-as-rotation ke liye, dekho Complex numbers — polar form and Euler's formula — parent e z = e x ( cos y + i sin y )
z ˉ = x − i y y → − y x
∣ z ∣ = x 2 + y 2 O z length . x y
Topic ko dono kyun chahiye: parent ka Example 2 z ˉ mirror — orientation-reversing, isliye accha nahi ), aur Example 4 ∣ z ∣ 2 = x 2 + y 2 forbid karne ki baat ka dil hai.
Intuition Ek map ko do heightmaps mein split karnaEk complex function har input point ( x , y ) u v kahan se shuru kiya , u v x y
f ( z ) = u ( x , y ) + i v ( x , y )
f → u ( x , y ) v ( x , y ) u v ( x , y ) i
Kyun: Cauchy–Riemann equations exactly in do terrains ke beech ek coupling hain. Inhe u v
Intuition "Partial" kyun?u do inputs par depend karta hai. Slope measure karne ke liye hume ek ko fix karna hoga aur doosre ko wiggle karna hoga — warna "the" slope ambiguous hai. Ek partial derivative ek-ko-chodkar-baaki-sab variable ko freeze karta hai.
Definition Partial derivativeu x = ∂ x ∂ u u y x Δ x u u x = lim Δ x → 0 Δ x u ( x + Δ x , y ) − u ( x , y ) . slice ka slope jo terrain ko x u y y
Kyun: CR equations u x = v y u y = − v x in chaar slopes ke beech relationships hain. Yeh topic ka punchline hai; yeh uske letters hain.
Yeh slopes kahan ek full local-linearisation matrix mein assemble hote hain, uske liye dekho Jacobian and the multivariable chain rule .
"lim Δ z → 0 ( stuff ) Δ z
Intuition Woh twist jo complex analysis ko mushkilaur sundar banata hai
Ek line par tum sirf left ya right se 0 Δ z = Δ x + i Δ y infinitely many directions se shrink ho sakta hai: x y same number par land karna chahiye.
f ′ ( z 0 ) f ′ ( z 0 ) = lim Δ z → 0 Δ z f ( z 0 + Δ z ) − f ( z 0 ) , jo ki har approach direction ke liye same value dene ki requirement hai. Yahi ek demand hai jo parent note do axes par squeeze karke Cauchy–Riemann equations nikalta hai.
Direction-independence kyun? Tabhi "Δ z Δ z f ′
Definition Parts match karnaDo complex numbers equal hote hain sirf tab jab unke real parts agree karein aur unke imaginary parts agree karein — woh ek mein do alag equations chhupee hoti hain. Picture: do points tabhi coincide karte hain jab unka right-coordinate AUR up-coordinate dono same ho.
Kyun: Parent ke Step 3 mein u x + i v x = v y − i u y split kiya jaata hai. "Matching parts" ke bina woh split ek mystery hai.
Imaginary unit i quarter turn
Complex number z = x + iy
Function f = u + i v two terrains
Partial derivatives u_x u_y v_x v_y
Limit and direction independence
Complex derivative f prime of z
Analytic functions the topic
Self-test: daayein side cover karo aur reveal karne se pehle har answer do.
i karta hai?Use origin ke around 90° counter-clockwise rotate karta hai (i 2 = − 1
z = x + i y y Ek ordinary real number — i Im ( z ) = y i y
z ˉ Use real axis ke across mirror karta hai (y → − y
∣ z ∣ Origin se
z tak ki distance, yaani arrow ki length,
x 2 + y 2 .
Hum f u v Output ek 2D point hai; u v ( x , y )
u x y x u
Ek complex limit real limit se zyada strict kyun hai? Wiggle Δ z 0
Woh kaunsa rule hai jis se ek complex equation do real equations ban jaati hai? Do complex numbers equal hote hain iff unke real parts match karein AUR unke imaginary parts match karein.
