22.Complex Analysis:Cauchy Integral Formula|UPSC|CSE|IFoS|CSIR NET|JAM|Shivraj Gurjar, IIT Kharagpur

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In this video , Cauchy Integral Formula is explained with illustrations.
For other Complex Analysis lectures , watch below linked playlist:-

Complex Analysis (सम्मिश्र विश्‍लेषण) is one of the important chapters in CSE and Indian forest Service Examination mathematics optional.It is also important for IIT JAM, GATE, CSIR NET (Mathematical Sciences), State Forest Services, ACF Exams, State PSCs etc.

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Hello Sir, at 2:16 can we conclude that {f(z)-f(z0)}/(z-z0) is f'(z) and since f(z) is analytic hence f'(z) will also be analytic in the domain. Following this, integration of f'(z)dz over closed curve will be zero (using CFT) and hence the result. Here I am using directly CFT instead of proving the middle part as zero using limit approach

kamaleshkumaryadav
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Sir in the end part of proof of cauchy integral formula, if we are taking epsilon as the radius that tends to zero, where z-z0= Ee^i theta then arent we just proving it in the neighbourhood of the point z0? How did we apply it to a bigger curve C inside the domain ?? Can you please elaborate on this sir? 6:46

anmolagarwal
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agar cauchy exterior theorem se Example ko krein to answer 0 ana chahiye??

onlyiit
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Sir in this if z_not lies inside the domain then only we appy this integral formula right? Otherwise the integral is 0 only?

namanbhatia
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Thank you sir, thank you so much 😭😭😭😭

mridulvatsalsingh
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Sir what if f(z) is not analytic on the bdry curve C? Will CIF still be valid?
My answer:
I think if f has a singularity on C then we cannot apply Cauchy Goursat to use continuity and say that the formula is valid.
So CIF is invalid.
Only on circles that f is analytic on, can be deformed to smaller circles by Cauchy Goursat and the formula will be valid.
Please correct me sir.

krishansai