What is a Contour? Complex Analysis Explained

[aaaa202]

Simple question, this word keeps popping up in complex analysis but my book hasn't given an exact definition of what a contour is. Is it just a region in the complex plane bounded by a curve? So when the books says for instance "integrate over a contour" is that equivalent to just integrating over a closed surface in ℝ²?

Another question which also troubles me quite a bit:

Due to the cauchy riemann relations we have:

\oint_Cf(z)dz = 0

But still I often see things like: "Consider two closed contours A and B in the argand diagram, B being sufficiently small that it lies completely within C. Show that if f(z) is analytic in the region between the two contours then:
\oint_Cf(z)dz = \oint_Bf(z)dz
But shouldn't that be obvious since they are both equal to zero?

 

[LCKurtz]

Simple question, this word keeps popping up in complex analysis but my book hasn't given an exact definition of what a contour is. Is it just a region in the complex plane bounded by a curve? So when the books says for instance "integrate over a contour" is that equivalent to just integrating over a closed surface in ℝ²?

Another question which also troubles me quite a bit:

Due to the cauchy riemann relations we have:

\oint_Cf(z)dz = 0

But still I often see things like: "Consider two closed contours A and B in the argand diagram, B being sufficiently small that it lies completely within C. Show that if f(z) is analytic in the region between the two contours then:
\oint_Cf(z)dz = \oint_Bf(z)dz
But shouldn't that be obvious since they are both equal to zero?

The contour integral is zero if f(z) is analytic inside the contour. But suppose f has a pole inside the contour. That is the situation being considered in your last example. Both contour integrals will give the same non-zero answer.

 

[Ray Vickson]

Simple question, this word keeps popping up in complex analysis but my book hasn't given an exact definition of what a contour is. Is it just a region in the complex plane bounded by a curve? So when the books says for instance "integrate over a contour" is that equivalent to just integrating over a closed surface in ℝ²?

Another question which also troubles me quite a bit:

Due to the cauchy riemann relations we have:

\oint_Cf(z)dz = 0

But still I often see things like: "Consider two closed contours A and B in the argand diagram, B being sufficiently small that it lies completely within C. Show that if f(z) is analytic in the region between the two contours then:
\oint_Cf(z)dz = \oint_Bf(z)dz
But shouldn't that be obvious since they are both equal to zero?

Contours are curves, not regions. Typically, a "contour integral" refers to an integral over a closed curve.

RGV