Contour integration
For continuous functions

To define the contour integral in this way one must first consider the integral, over a real variable, of a complex-valued function. Let f: R → C {\displaystyle f:\mathbb {R} \to \mathbb {C} } be a complex-valued function of a real variable, t {\displaystyle t}. The real and imaginary parts of f {\displaystyle f} are often denoted as u (t) {\displaystyle u(t)} and v (t) {\displaystyle v(t)}, respectively, so that f (t) = u (t) + i v (t). {\displaystyle f(t)=u(t)+iv(t).} Then the integral of the complex-valued function f {\displaystyle f} over the interval {\displaystyle } is given by ∫ a b f (t) d t = ∫ a b (u (t) + i v (t)) d t = ∫ a b u (t) d t + i ∫ a b v (t) d t. {\displaystyle {\begin{aligned}\int _{a}^{b}f(t)\,dt&=\int _{a}^{b}{\big (}u(t)+iv(t){\big)}\,dt\\&=\int _{a}^{b}u(t)\,dt+i\int _{a}^{b}v(t)\,dt.\end{aligned}}}

Now, to define the contour integral, let f: C → C {\displaystyle f:\mathbb {C} \to \mathbb {C} } be a continuous function on the directed smooth curve γ {\displaystyle \gamma }. Let z: → C {\displaystyle z:\to \mathbb {C} } be any parametrization of γ {\displaystyle \gamma } that is consistent with its order (direction). Then the integral along γ {\displaystyle \gamma } is denoted ∫ γ f (z) d z {\displaystyle \int _{\gamma }f(z)\,dz\,} and is given by ∫ γ f (z) d z:= ∫ a b f (z (t)) z ′ (t) d t. {\displaystyle \int _{\gamma }f(z)\,dz:=\int _{a}^{b}f{\big (}z(t){\big)}z'(t)\,dt.}

This definition is well defined. That is, the result is independent of the parametrization chosen. In the case where the real integral on the right side does not exist the integral along γ {\displaystyle \gamma } is said not to exist.
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