Difference between revisions of "Physical / geometric interpretation of complex integration"
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There are many geometric and physical interpretations of integrals of real-valued functions (such as are, volume, work etc.). However, a similar interpretation is rarely discussed for complex contour integral. The following interpretation of contour integration is due to Pólya (George Pólya and Gordon Latta, Complex Variables, Wiley, 1974.) | There are many geometric and physical interpretations of integrals of real-valued functions (such as are, volume, work etc.). However, a similar interpretation is rarely discussed for complex contour integral. The following interpretation of contour integration is due to Pólya (George Pólya and Gordon Latta, Complex Variables, Wiley, 1974.) | ||
| − | Let <math> \gamma </math> be a simple closed smooth curve in the complex plane, and <math> f </math> be a (complex-valued) function defined on <math> \gamma </math>. Parametrize <math> \gamma </math> with arc length <math> s </math>. As usual, have <math> f(z) = u(z) + iv(z) </math>. Consider a vector field | + | Let <math> \gamma </math> be a simple closed smooth curve in the complex plane, and <math> f </math> be a (complex-valued) function defined on <math> \gamma </math>. Parametrize <math> \gamma </math> with arc length <math> s </math>, such that <math> \gamma(s) = x(s) + y(s). </math> As usual, have <math> f(z) = u(z) + iv(z) = u(x,y) + iv(x,y)</math>. Consider a vector field |
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| + | <math> {\bf F} (x,y) = [u(x,y), -v(x,y)]. </math> | ||
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| + | We have: | ||
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| + | <math> \int _\gamma f(z)\, dz = \int_0^l f(\gamma(s) ) \gamma' (s) \, ds = \int_0^l (u(\gamma(s) +i v(\gamma(s))(x'(s) + i y'(s)) \, ds= </math> | ||
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| + | <math> = \int_0^l \left (u(\gamma(s))x'(s) - v(\gamma(s)) y'(s)\right) \, ds + i \int_0^l \left (u(\gamma(s))y'(s) + v(\gamma(s)) x'(s)\right) \, ds =</math> | ||
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| + | <math> = \int_0^l [u(\gamma(s), -v(\gamma(s)] \cdot [x'(s), y'(s)] \, ds + i \int_0^l [u(\gamma(s)), -v(\gamma(s)]\cdot [y'(s), -x'(s)] \, ds =</math> | ||
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| + | <math> = \int_0^l {\bf F}(\gamma(s)) \cdot {\bf T}(\gamma (s)) \, ds + i \int_0^l {\bf F}(\gamma (s))\cdot {\bf N}(\gamma (s)) \, ds =</math> | ||
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| + | <math> = \int_\gamma {\bf F } \cdot {\bf T}\, d\gamma + i \int_\gamma \bf{F} \cdot {\bf N}\, d\gamma,</math> | ||
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| + | where <math> {\bf T} </math> and <math> {\bf N} </math> are, respectively, the unit tangent and the unit normal vectors to the curve <math> \gamma.</math> | ||
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| + | This computation shows that <math> {\rm Re}\left(\int _\gamma f(z)\, dz\right) </math> can be interpreted as the flow of the vector field <math> {\bf F} </math> along <math> \gamma </math>, while <math> {\rm Im}\left(\int _\gamma f(z)\, dz\right) </math> can be interpreted as the flux of <math> {\bf F} </math> across <math> \gamma </math>. | ||
Latest revision as of 11:23, 21 December 2021
There are many geometric and physical interpretations of integrals of real-valued functions (such as are, volume, work etc.). However, a similar interpretation is rarely discussed for complex contour integral. The following interpretation of contour integration is due to Pólya (George Pólya and Gordon Latta, Complex Variables, Wiley, 1974.)
Let [math]\displaystyle{ \gamma }[/math] be a simple closed smooth curve in the complex plane, and [math]\displaystyle{ f }[/math] be a (complex-valued) function defined on [math]\displaystyle{ \gamma }[/math]. Parametrize [math]\displaystyle{ \gamma }[/math] with arc length [math]\displaystyle{ s }[/math], such that [math]\displaystyle{ \gamma(s) = x(s) + y(s). }[/math] As usual, have [math]\displaystyle{ f(z) = u(z) + iv(z) = u(x,y) + iv(x,y) }[/math]. Consider a vector field
[math]\displaystyle{ {\bf F} (x,y) = [u(x,y), -v(x,y)]. }[/math]
We have:
[math]\displaystyle{ \int _\gamma f(z)\, dz = \int_0^l f(\gamma(s) ) \gamma' (s) \, ds = \int_0^l (u(\gamma(s) +i v(\gamma(s))(x'(s) + i y'(s)) \, ds= }[/math]
[math]\displaystyle{ = \int_0^l \left (u(\gamma(s))x'(s) - v(\gamma(s)) y'(s)\right) \, ds + i \int_0^l \left (u(\gamma(s))y'(s) + v(\gamma(s)) x'(s)\right) \, ds = }[/math]
[math]\displaystyle{ = \int_0^l [u(\gamma(s), -v(\gamma(s)] \cdot [x'(s), y'(s)] \, ds + i \int_0^l [u(\gamma(s)), -v(\gamma(s)]\cdot [y'(s), -x'(s)] \, ds = }[/math]
[math]\displaystyle{ = \int_0^l {\bf F}(\gamma(s)) \cdot {\bf T}(\gamma (s)) \, ds + i \int_0^l {\bf F}(\gamma (s))\cdot {\bf N}(\gamma (s)) \, ds = }[/math]
[math]\displaystyle{ = \int_\gamma {\bf F } \cdot {\bf T}\, d\gamma + i \int_\gamma \bf{F} \cdot {\bf N}\, d\gamma, }[/math]
where [math]\displaystyle{ {\bf T} }[/math] and [math]\displaystyle{ {\bf N} }[/math] are, respectively, the unit tangent and the unit normal vectors to the curve [math]\displaystyle{ \gamma. }[/math]
This computation shows that [math]\displaystyle{ {\rm Re}\left(\int _\gamma f(z)\, dz\right) }[/math] can be interpreted as the flow of the vector field [math]\displaystyle{ {\bf F} }[/math] along [math]\displaystyle{ \gamma }[/math], while [math]\displaystyle{ {\rm Im}\left(\int _\gamma f(z)\, dz\right) }[/math] can be interpreted as the flux of [math]\displaystyle{ {\bf F} }[/math] across [math]\displaystyle{ \gamma }[/math].