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4.2: Complex Line Integrals - Mathematics LibreTexts

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Line integrals are also called path or contour integrals. Given the ingredients we define the complex lineintegral ∫ γ f(z) dz ∫ 𝛾 𝑓 ( 𝑧 ) 𝑑 𝑧 by ∫ γ f(z) dz:= ∫ b a f(γ(t)) γ ′ (t) dt. (4.2.1) (4.2.1) ∫ 𝛾 𝑓 ( 𝑧 ) 𝑑 𝑧 := ∫ 𝑎 𝑏 𝑓 ( 𝛾 ( 𝑡 ) ) 𝛾 ′ ( 𝑡 ) 𝑑 𝑡 . You should note that this notation looks just like integrals of a real variable. We don’t need the vectors and dot products of line integrals in R 2 𝑅 2 . Also, make sure you understand that the product f(γ(t)) γ ′ (t) 𝑓 ( 𝛾 ( 𝑡 ) ) 𝛾 ′ ( 𝑡 ) is just a product of complex numbers. An alternative notation uses dz=dx+idy 𝑑 𝑧 = 𝑑 𝑥 + 𝑖 𝑑 𝑦 to write ∫ γ f(z) dz= ∫ γ (u+iv)(dx+idy) (4.2.2) (4.2.2) ∫ 𝛾 𝑓 ( 𝑧 ) 𝑑 𝑧 = ∫ 𝛾 ( 𝑢 + 𝑖 𝑣 ) ( 𝑑 𝑥 + 𝑖 𝑑 𝑦 ) Let’s check that Equations 4.2.1 4.2.1 and 4.2.2 4.2.2 are the same. Equation 4.2.2 4.2.2 is really a multivariable calculus expression, so thinking of γ(t) 𝛾 ( 𝑡 ) as (x(t),y(t)) ( 𝑥 ( 𝑡 ) , 𝑦 ( 𝑡 ) ) it becomes ∫ γ f(z)

Line integrals are also called path or contour integrals. Given the ingredients we define the complex lineintegral \(\int_{\gamma} f(z)\ dz\) by \[\int_{\gamma} f(z)\ dz := \int_{a}^{b} f(\gamma (t)) \gamma ' (t)\ dt. \label{4.2.1} \] You should note that this notation looks just like integrals of a real variable. We don't need the vectors and dot products of line integrals in \(R^2\). Also, make sure you understand that the product \(f(\gamma (t)) \gamma '(t)\) is just a product of complex numbers. An alternative notation uses \(dz = dx + idy\) to write \[\int_{\gamma} f(z)\ dz = \int_{\gamma

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