Evaluate the integral ∫𑀾 𝑧̅ 𝑑𝑧 where 𝐶 is the straight line from the point (1,0) to the point (1,1)

𝑪𝒐𝒎𝒑𝒍𝒆𝒙 𝑨𝒏𝒂𝒍𝒚𝒔𝒊𝒔 𝑻𝒉𝒆𝒐𝒓𝒆𝒎 𝒇𝒓𝒐𝒎 𝑪𝒐𝒎𝒑𝒍𝒆𝒙 𝑰𝒏𝒕𝒆𝒈𝒓𝒂𝒍

𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏:-

Evaluate the integral ∫𑀾 𝑧̅ 𝑑𝑧 , where 𝐶 is the straight line from the point (1,0) to the point (1,1) .

𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:-

We know that 𝑧 = 𝑥+𝑖𝑦
∴ 𝑑𝑧=𝑑𝑥+𝑖𝑑𝑦 ⇒𝑑𝑧=𝑖𝑑𝑦 (∵𝑥=1⇒𝑑𝑥=0)
and 𝑧̅ = 𝑥 - 𝑖𝑦
Here line 𝐴𝐵 , 𝑥=1 and 𝑦 moves 0 to 1 .
∫𑀾 𝑧̅ 𝑑𝑧 = ∫ ᤲ ᷝ(1-𝑖𝑦)𝑖𝑑𝑦
⇒∫𑀾 𝑧̅ 𝑑𝑧 = 𝑖∫ ᤲ ᷝ(1-𝑖𝑦)𝑑𝑦
⇒∫𑀾 𝑧̅ 𝑑𝑧 = 𝑖[ 𝑦] ᤲ ᷝ+[ 𝑦²/2] ᤲᤲᤲ ᷝ
⇒∫𑀾 𝑧̅ 𝑑𝑧 = 𝑖[1-0] + [(1/2)-0]
⇒∫𑀾 𝑧̅ 𝑑𝑧 = 𝑖+(1/2)
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Solved...!!!
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#complexanalysis #csirnet #complexintegration #riemannintegral
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