Evaluate ∫𑀾 𝑧̅ 𝑑𝑧 , from 𝑧 = 0 to 𝑧 = 4+2𝑖 along the curve 𝐶 given by 𝑧 = 𝑡² + 𝑖𝑡 .

Показать описание

𝑪𝒐𝒎𝒑𝒍𝒆𝒙 𝑨𝒏𝒂𝒍𝒚𝒔𝒊𝒔 𝑻𝒉𝒆𝒐𝒓𝒆𝒎 𝒇𝒓𝒐𝒎 𝑪𝒐𝒎𝒑𝒍𝒆𝒙 𝑰𝒏𝒕𝒆𝒈𝒓𝒂𝒍

𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏:-

Evaluate ∫𑀾 𝑧̅ 𝑑𝑧 , from 𝑧 = 0 to 𝑧 = 4+2𝑖 along the curve 𝐶 given by 𝑧 = 𝑡²+𝑖𝑡 .

𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:-

Given that the curve
𝑧 = 𝑡²+𝑖𝑡
⇒ 𝑥+𝑖𝑦 = 𝑡²+𝑖𝑡
⇒ 𝑥 = 𝑡² and 𝑦 = 𝑡
⇒ 𝑥 = 𝑦² this is the equation of parabola for the given parametric equation 𝑧 = 𝑡²+𝑖𝑡
∴ 𝑧 → 0 ⇒ 𝑡 → 0 and 𝑧 → 4+2𝑖 ⇒ 𝑡 → 2
and we know that 𝑧 = 𝑡²+𝑖𝑡 ∴ 𝑑𝑧=(2𝑡+𝑖)𝑑𝑡
and 𝑧̅ = 𝑡² - 𝑖𝑡
Now Putting in the given integral

∫𑀾 𝑧̅ 𝑑𝑧 = ∫꣢ᤲ(𝑡²+𝑖𝑡)(2𝑡+𝑖)𝑑𝑡
⇒∫𑀾 𝑧̅ 𝑑𝑧 = ∫꣢ᤲ(2𝑡³+𝑡-𝑖𝑡²)𝑑𝑡
⇒∫𑀾 𝑧̅ 𝑑𝑧 = [2(𝑡⁴/4)+(𝑡²/2)]꣢ᤲ - 𝑖 [(𝑡³/3)]꣢ᤲ
⇒∫𑀾 𝑧̅ 𝑑𝑧 = [2(16/4)+(4/2) - 0 - 0] - 𝑖 [(8/3) - 0]
⇒∫𑀾 𝑧̅ 𝑑𝑧 = [2(16/4)+(4/2) - 0 - 0] - 𝑖 [(8/3) - 0]
⇒∫𑀾 𝑧̅ 𝑑𝑧 = 10 - (8/3)𝑖
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Solved...!!!
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#complexanalysis #csirnet #complexintegration #riemannintegral
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