Complex Line Integrals

(6.158) |

Because , and writing , where and are real functions, it follows that , where

(6.159) | ||

(6.160) |

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

(6.161) | ||

(6.162) |

where , , , , and , . According to the curl theorem (see Section A.22),

(6.163) | ||

(6.164) |

where is the region of the - plane enclosed by . Hence, we obtain

(6.165) | ||

(6.166) |

Let

(6.167) | ||

(6.168) |

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

(6.169) |

Writing , a direct generalization of the previous analysis reveals that

(6.170) | ||

(6.171) |

where is now the region of the - plane lying between the curves and . Suppose that is well-behaved (i.e., finite, single-valued, and differentiable) throughout . It immediately follows that its real and imaginary components, and , respectively, satisfy the Cauchy-Riemann relations, (6.17)-(6.18), throughout . However, if this is the case then it is apparent, from the previous two expressions, that . In other words, if is well-behaved throughout then .

The circulation of the flow about some closed curve in the - plane is defined

where 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 -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 while still remaining in the fluid, and not passing over a singularity of the complex velocity, .