# What is a contour?

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
Homework Helper
Gold Member
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
Homework Helper
Dearly Missed
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