Evaluate the integral 𝐼=∫𑀾 𝑅𝑒𝑧 𝑑𝑧 , on the circle |𝑧|=𝑟 .

𝑪𝒐𝒎𝒑𝒍𝒆𝒙 𝑨𝒏𝒂𝒍𝒚𝒔𝒊𝒔 𝑻𝒉𝒆𝒐𝒓𝒆𝒎 𝒇𝒓𝒐𝒎 𝑪𝒐𝒎𝒑𝒍𝒆𝒙 𝑰𝒏𝒕𝒆𝒈𝒓𝒂𝒍

𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏:-

Evaluate the integral 𝐼=∫𑀾 𝑅𝑒𝑧 𝑑𝑧 , on the circle |𝑧|=𝑟 .

𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:-

We know that 𝑅𝑒𝑧 = ( 𝑧+ 𝑧̅ )/2
∵ 𝑧 = 𝑟𝑒ⁱᶿ and 𝑧̅ = 𝑟𝑒⁻ⁱᶿ
∴𝑅𝑒𝑧 = 𝑟(𝑒ⁱᶿ+ 𝑒⁻ⁱᶿ)/2
Now 𝑧 = 𝑟𝑒ⁱᶿ⇒ 𝑑𝑧 = 𝑖𝑟𝑒ⁱᶿ𝑑𝜽

We have the integration

𝐼=∫𑀾 𝑅𝑒𝑧 𝑑𝑧
⇒ 𝐼=∫𑀾 {𝑟(𝑒ⁱᶿ+ 𝑒⁻ⁱᶿ)/2}𝑖𝑟𝑒ⁱᶿ𝑑𝜽
⇒2𝐼=𝑖𝑟²∫𑀾 𝑒ⁱᶿ(𝑒ⁱᶿ+ 𝑒⁻ⁱᶿ)𝑑𝜽
⇒2𝐼=𝑖𝑟²∫𑀾 (𝑒ⁱ²ᶿ+ 1)𝑑𝜽
⇒2𝐼=𝑖𝑟²∫𑀾 (𝑐𝑜𝑠2𝜽+𝑖𝑠𝑖𝑛2𝜽+ 1)𝑑𝜽

Here limits are 0→2𝝅 for Circle

⇒2𝐼=𝑖𝑟²[(𝑠𝑖𝑛2𝜽/2)-𝑖(𝑐𝑜𝑠2𝜽/2)+𝜽]² ᤲⷧ
⇒2𝐼=𝑖𝑟²[(0-0)-𝑖(1-1)+2𝝅-0]
⇒2𝐼=𝑖2𝝅𝑟²
⇒𝐼=𝑖𝝅𝑟²
------------------------------------------------------------------
.
.
.
#complexanalysis #csirnet #complexintegration #riemannintegral
.

🔥🔥🔥Hurry Up Students for whole detailed Videos with Answer in Description for making notes 🔥🔥🔥

🔥🔥🔥JOIN JOIN JOIN JOIN 🔥🔥🔥

🔥🔥🔥All B.Sc., B.Tech , M.Sc. , M.A. Mathematics students join with us A great platform to get knowledge in Mathematics and science... Stay tuned....for detailed videos on whole SYLLABUS with topic wise...and also join us for CSIR-UGC National Eligibility Test (NET) for Junior Research Fellowship and Lecturer-ship COMMON SYLLABUS FOR PART 'B' AND 'C' MATHEMATICAL SCIENCE 🔥🔥🔥
------------------------------------------------------------------
Channel Link: 🔥🔥🔥

Videos links 🔥🔥🔥
👇👇👇👇
------------------------------------------------------------------

̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅
Play List ( Complex Analysis )

------------------------------------------------------------------
------------------------------------------------------------------
Kindly join Telegram channel 🔥🔥🔥

-------------------------------------------------------------