Complex Line Integrals

Consider the line integral of some function $F(z)$ of the complex variable taken (counter-clockwise) around a closed curve $C$ in the complex plane:

$$J = \oint_C F(z)\,dz.$$ (6.158)

Because $dz=dx+{\rm i}\,dy$ , and writing $F(z)=\phi(x,y)+{\rm i}\,\psi(x,y)$ , where $\phi(x,y)$ and $\psi(x,y)$ are real functions, it follows that $J=J_r+{\rm i}\,J_i$ , where

$$J_r =\oint_C (\phi\,dx-\psi\,dy),$$ (6.159)

$$J_i =\oint_C (\psi\,dx+\phi\,dy).$$ (6.160)

However, we can also write the previous expressions in the two-dimensional vector form

$$J_r =\oint_C {\bf A}\cdot d{\bf r},$$ (6.161)

$$J_i =\oint_C {\bf B}\cdot d{\bf r},$$ (6.162)

where $d{\bf r}=(dx$ , $dy)$ , ${\bf A}=(\phi$ , $-\psi)$ , and ${\bf B} = (\psi$ , $\phi)$ . According to the curl theorem (see Section A.22),

$$\oint_C {\bf A}\cdot d{\bf r} =\int_S (\nabla\times {\bf A})_z\,dS,$$ (6.163)

$$\oint_C {\bf B}\cdot d{\bf r} =\int_S (\nabla\times {\bf B})_z\,dS,$$ (6.164)

where $S$ is the region of the $x$ -$y$ plane enclosed by $C$ . Hence, we obtain

$$J_r =-\int_S\left(\frac{\partial\psi}{\partial x} + \frac{\partial\phi}{\partial y}\right)dS,$$ (6.165)

$$J_i =\int_S\left(\frac{\partial\phi}{\partial x} - \frac{\partial\psi}{\partial y}\right)dS.$$ (6.166)

Let

$$J =\oint_{C}F(z)\,dz,$$ (6.167)

$$J' =\oint_{C'} F(z)\,dz,$$ (6.168)

where $C'$ is a closed curve in the complex plane that completely surrounds the smaller curve $C$ . Consider

$${\mit\Delta} J = J-J'.$$ (6.169)

Writing ${\mit\Delta} J={\mit\Delta} J_r+{\rm i}\,{\mit\Delta} J_i$ , a direct generalization of the previous analysis reveals that

$${\mit\Delta} J_r =-\int_{S}\left(\frac{\partial\psi}{\partial x} + \frac{\partial\phi}{\partial y}\right)dS,$$ (6.170)

$${\mit\Delta} J_i =\int_{S}\left(\frac{\partial\phi}{\partial x} - \frac{\partial\psi}{\partial y}\right)dS,$$ (6.171)

where $S$ is now the region of the $x$ -$y$ plane lying between the curves $C$ and $C'$ . Suppose that $F(z)$ is well-behaved (i.e., finite, single-valued, and differentiable) throughout $S$ . It immediately follows that its real and imaginary components, $\phi$ and $\psi$ , respectively, satisfy the Cauchy-Riemann relations, (6.17)-(6.18), throughout $S$ . However, if this is the case then it is apparent, from the previous two expressions, that ${\mit\Delta} J_r={\mit\Delta} J_i=0$ . In other words, if $F(z)$ is well-behaved throughout $S$ then $J=J'$ .

The circulation of the flow about some closed curve $C$ in the $x$ -$y$ plane is defined

$${\mit\Gamma} = \oint_C (v_x\,dx+v_y\,dy) = -{\rm Re}\oint_C \frac{dF}{dz}\,dz,$$ (6.172)

where $F(z)$ is the complex velocity potential of the flow, and use has been made of Equation (6.35). Thus, the circulation can be evaluated by performing a line integral in the complex $z$ -plane. Moreover, as is clear from the previous discussion, this integral can be performed around any loop that can be continuously deformed into the loop $C$ while still remaining in the fluid, and not passing over a singularity of the complex velocity, $dF/dz$ .