Evaluate ∫𑀾 𝑑𝑧 in Complex Integration.

𝑪𝒐𝒎𝒑𝒍𝒆𝒙 𝑨𝒏𝒂𝒍𝒚𝒔𝒊𝒔 𝑻𝒉𝒆𝒐𝒓𝒆𝒎 𝒇𝒓𝒐𝒎 𝑪𝒐𝒎𝒑𝒍𝒆𝒙 𝑰𝒏𝒕𝒆𝒈𝒓𝒂𝒍

𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏:-

Evaluate ∫𑀾 𝑑𝑧 .

𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:-

According to Riemann's Definition of Integration -

∫𑀾 𝑓(𝑧)𝑑𝑧= 𝐥𝐢𝐦ₙ ͢ ͚Σᵣ₌ᷠ₁ 𝑓 (𝑒ᵣ)(𝑧ᵣ - 𝑧ᵣ₋₁)

Put 𝑓(𝑧)=1 ⇒ 𝑓 (𝑒ᵣ)=1 we get

⇒∫𑀾 𝑑𝑧= 𝐥𝐢𝐦ₙ ͢ ͚Σᵣ₌ᷠ₁ (𝑧ᵣ - 𝑧ᵣ₋₁)
⇒∫𑀾 𝑑𝑧= 𝐥𝐢𝐦ₙ ͢ ͚{(𝑧₁ - 𝑧₀)+(𝑧₂ - 𝑧₁)+.....+(𝑧ₙ - 𝑧ₙ₋₁)}
⇒∫𑀾 𝑑𝑧= 𝐥𝐢𝐦ₙ ͢ ͚(𝑧ₙ - 𝑧₀)
∵ 𝑧ₙ=𝑏 and 𝑧₀=𝑎

⇒∫𑀾 𝑑𝑧= 𝑏 - 𝑎

For closed curve we know that points 𝑎 and 𝑏 coincide ∴ 𝑏 = 𝑎

⇒∫𑀾 𝑑𝑧= 𝑏 - 𝑏

⇒∫𑀾 𝑑𝑧= 0 ( For closed curve )

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Solved....!!!
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#complexanalysis #csirnet #complexintegration #riemannintegral
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