Calculus Small Changes, Big Ideas

Section 19.3 Contour integrals

• A contour integral is just a line integral over a curve \(C\) in the complex plane:

  \begin{equation*} \int_{C} f(z)\dd{z} \end{equation*}

  We evaluate it by parametrizing \(C\text{:}\)

  \begin{equation*} \int_{a}^{b} f(z(t))z'(t)\dd{t} \end{equation*}

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• Great example to go through: if \(C\) is the unit circle parametrized counterclockwise, and \(n\neq-1\) is an integer, then

  \begin{equation*} \int_{C} z^{n}\dd{z}=0\text{.} \end{equation*}

  However, as an exception,

  \begin{equation*} \int_{C} \frac{1}{z}\dd{z}=2\pii\ii\text{.} \end{equation*}

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• Theorem 10. ML Inequality.

  If \(f\) is continuous on a smooth curve \(C\text{,}\) and if \(|f(z)|\le M\) for all \(z\) on \(C\text{,}\) then

  \begin{equation*} \abs{\int_C f(z)\dd{z}}\le ML\text{,} \end{equation*}

  where \(L\) is the length of \(C\text{.}\)

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• Suppose that \(\bv F(x,y)=(P(x,y),Q(x,y))\) is the Pólya vector field of the function \(f(z)=P(x,y)-\ii Q(x,y)\text{.}\) Then:

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  \begin{align*} \oint_C f(z)\dd{z} \amp = \oint_C (P-\ii Q)(\dd{x}+\ii\dd{y})\\ \amp = \qty(\oint_C P\dd{x}+Q\dd{y})+\ii\qty(\oint_C P\dd{y}-Q\dd{x})\\ \amp = \qty(\oint_C\bv F\cdot\uv{t}\dd{s})+\ii\qty(\oint_C\bv F\cdot\uv{n}\dd{s}) \end{align*}
  Thus the contour integral computes BOTH the circulation AND the flux at the same time!

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• Winding number?

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• Blasius’s theorem: The force experienced by a two-dimensional fixed body in a steady irrotational flow is given by

  \begin{equation*} L-\ii D=\frac{\ii\rho}{2}\oint_C \qty(\dv{w}{z})^2\dd{z}\text{,} \end{equation*}

  where \(w(z)=\phi(x,y)+\ii\,\psi(x,y)\) is the complex potential of the flow, \(C\) is a contour enclosing the body, and \(\rho\) is the fluid density. Here, \(L\) is the lift and \(D\) is the drag. (Note the conjugate.)

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  Furthermore, the moment about the origin is given by

  \begin{equation*} M=\RE\qty{-\frac{\rho}{2}\oint_C z\qty(\dv{w}{z})^2\dd{z}}\text{.} \end{equation*}

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