4.10.3 · D5 · HinglishAdvanced Topics (Elite Level)
Question bank — Cauchy's integral theorem and formula
4.10.3 · D5· Maths › Advanced Topics (Elite Level) › Cauchy's integral theorem and formula
Neeche use hone wale notation ke reminders, taaki kuch unexplained na rahe:
- ka matlab hai "closed loop ke around ek baar chalte waqt ko add karo." Ek loop closed hota hai jab wo apne starting point pe wapas aata hai.
- Holomorphic = complex-differentiable (har direction se same limit) — parent mein Cauchy-Riemann equations se build hota hai.
- Simply connected = "koi hole nahi": har loop region ke andar ek point tak shrink ho sakti hai.
- pe ek pole wo point hai jahan , ki tarah blow up karta hai.
- Positively oriented ka matlab hai loop counter-clockwise traverse hoti hai, taaki enclosed region tumhare left pe rahe.
True or false — justify karo
Har line is format mein hai: statement ::: verdict + asli reason.
har holomorphic aur har closed loop ke liye.
False — sirf tabhi jab koi singularity enclose na kare aur poore enclosed region pe holomorphic ho. ke around ka integral deta hai, nahi.
Agar poore mein everywhere holomorphic (entire) hai, toh kisi bhi closed ke liye .
True — ek entire function mein kahin koi singularity nahi hoti, isliye har enclosed region singularity-free hai aur Cauchy's theorem apply hota hai.
aur ek hi value dete hain.
True — dono loops pe single pole enclose karti hain; ko tak deform karne mein koi nayi singularity cross nahi hoti, isliye integral unchanged rehta hai.
ki orientation ulat dene se (counter-clockwise ki jagah clockwise) unchanged rehta hai.
False — direction reverse karne se sign flip ho jaata hai: . Cauchy's formula deta.
ki value ki exact shape pe depend karti hai.
False — ye sirf is baat pe depend karta hai ki andar hai ya nahi; koi bhi do loops jo ko enclose karti hain (aur koi doosri singularity nahi) wahi deti hain.
Agar kisi ek particular loop ke liye hai, toh andar holomorphic honi chahiye.
False — integral accidentally zero ho sakta hai (jaise symmetric cancellation se) singularity hone ke bawajood; ek single zero holomorphy certify nahi karta.
Jo function ek open set pe ek baar bhi complex-differentiable hai, wo automatically wahan infinitely differentiable hai.
True — generalized formula har ko ke ek contour integral ke roop mein likhta hai, jo sabhi ke liye exist karta hai. Real variables mein iska koi analogue nahi.
Cauchy's theorem kisi bhi holomorphic ke liye ek annulus (hole wala disk) pe hold karta hai.
False — ek annulus simply connected nahi hota; hole ke around ek loop ek point tak shrink nahi ho sakti, isliye nonzero ho sakta hai chahe annulus pe holomorphic ho.
Error dhundho
Har line ek flawed argument quote karta hai; reveal mein flaw ka naam bataya gaya hai.
" polynomials ka ratio hai, isliye ye holomorphic hai, isliye ."
Flaw: , pe holomorphic nahi hai, jo ke andar hai. Holomorphy poore enclosed region pe hold honi chahiye, sirf uske zyaadatar hisse pe nahi.
" integral formula se."
Flaw: , ke bahar hai. Andar koi pole enclosed nahi hai, integrand andar holomorphic hai, isliye answer hai, na ki .
" jahan ."
Flaw: pole order 3 ka hai, isliye tumhe ke saath generalized formula wala use karna hoga: answer hai, na ki .
" ke poles ke andar aur pe hain, isliye use karo ke saath."
Flaw: formula require karta hai ki (numerator) poore interior pe holomorphic ho; pe doosra pole ise violate karta hai. Partial fractions se split karo ya Residue theorem use karo.
", isliye constant hai."
Flaw: vanishing loop integral sirf path-independence/holomorphy signal karta hai, constancy nahi. Constancy ke liye ek global extra hypothesis chahiye — wo hai Liouville's bounded-entire argument.
"Kyunki ke interior mein ek pole hai, Green's theorem phir bhi mujhe integral ko pe collapse karne deta hai."
Flaw: Green's theorem demand karta hai ki integrand poore region pe smooth ho. Ek pole double integral ko improper/undefined bana deta hai, isliye Green's theorem collapse exactly wahan illegal hai.
" kyunki ye ek nice rational function hai."
Flaw: ke poles pe hain, dono ke andar. Integral do residue contributions ka sum hai, automatically nahi.
Why questions
Holomorphy ki "har direction se same limit" wali demand Cauchy's theorem kyun deti hai, jabki real differentiability nahi deti?
Directional agreement Cauchy-Riemann equations ko force karta hai, jo Green's-theorem ke dono integrands ko zero bana deta hai. Real differentiability sirf do directions check karti hai, aisi koi curl-free structure nahi milti.
"Simply connected" sirf cosmetic ki jagah essential kyun hai?
Green's theorem poore enclosed region pe integrate karta hai; ek hole (enclosed singularity) wo point hai jahan argument fail hota hai, aur yahi failure exactly banati hai.
Integral formula mein kyun aata hai?
Shrunken circle pe factors cancel ho jaate hain, jo chhod jaata hai. literally wo angle hai jo ek baar around sweep hota hai.
Hum ko ke around ek tiny circle tak shrink kyun kar sakte hain bina integral change kiye?
aur chhoti circle ke beech integrand holomorphic hai (wahan koi pole nahi), isliye Cauchy's theorem dono loop integrals ka difference zero bana deta hai.
Derivative formula mein kyun aata hai?
ko exactly baar differentiate karne se milta hai; wo factorial tab saath aata hai jab tum integral sign ke neeche baar differentiate karte ho.
ko sirf boundary pe jaanna ko andar everywhere kyun determine karta hai?
Formula sirf ki boundary values use karta hai. Holomorphy itni rigid hai ki interior forced ho jaata hai — yahi rigidity Maximum modulus principle ke neeche bhi hai.
Ek holomorphic function ki power series (uski Laurent series bina negative terms ke) converge kyun guarantee hoti hai?
Generalized formula har ko boundary integral se bound karta hai, jo Cauchy estimates deta hai jo Taylor coefficients ko itni tezi se shrink karne force karta hai ki convergence ho jaaye.
Edge cases
jab exactly pe lie karta hai (na andar, na bahar)?
Standard formula se undefined — integrand contour pe hi blow up karta hai. Koi principal value lena padta hai ya contour ko indent karna padta hai; clean ab hold nahi karta.
jab ek aisi loop hai jo ke around do baar wind karti hai?
Tumhe milta hai: value winding number se multiply hoti hai, jo count karta hai ki ke around kitne net counter-clockwise turns karta hai.
jab , ke bahar hai?
Zero — koi pole enclosed nahi hai, integrand poore interior pe holomorphic hai, isliye Cauchy's theorem apply hota hai.
Generalized formula jab ?
Ye ordinary integral formula mein collapse ho jaata hai, kyunki aur . Dono formulas ek hi family ke hain.
ek zero enclosed area wali loop pe (ek aisi contour jo khud pe double back karti hai, kuch bhi enclose nahi karti)?
Zero — ek degenerate loop jo koi region enclose nahi karti wo Green's theorem mein net area contribute nahi karti, isliye integral trivially vanish karta hai.
jab boundary point andar se?
ki value tab bhi hold karti hai jab strictly andar hai, lekin limit mein ye singular ho jaata hai — formula exactly tab degrade hota hai jab contour tak pahunchta hai.
ke andar do poles: kya Cauchy's formula seedha apply hota hai?
Nahi — formula exactly ek pole pick off karta hai. Kaafi enclosed poles ke saath partial fractions se split karo ya seedha Residue theorem pe jaao, jo saare enclosed residues sum karta hai.
poore pe holomorphic aur bounded: machinery kya force karti hai?
Derivative formula se Cauchy estimates drive kar dete hain, isliye constant hai — ye Liouville's theorem hai, integral formula ka direct child.
Recall Jaane se pehle ek-line self-test
Theorem kehta hai loop = ? ::: , lekin sirf tab jab koi enclosed singularity na ho. Formula kehta hai loop = ? ::: , enclosed value harvest karta hai. Wo ek cheez jo dono ko break karti hai jab violate hoti hai ::: ek enclosed singularity / simple connectivity ka loss.