How could contour integration be represented as a number?

Question

What is a contour integral? In what ways can contour integration be represented as a number?

Answer 1

A normal integral $ \int_a^b f(x) dx $ represents an "accumulation" of $ f $'s values over the interval $ (a,b) $; if the integral's result is divided by the length of the path $ b-a $ one ends up with the mean value of $f$ on that interval. A "weighted" integral $ \int_a^b f(x) w(x) dx $ accumulates $ f $'s values very similarly, but in this case it weighs $ f $ differently at every point using what's called the weight function $ w(x) $. A path integral $ \int_\gamma f(z) dz $ represents almost the very same idea: it accumulates the complex values of $ f $ over the points of a path, but "weighed" according to the tangent direction (interpreted as a complex number) of the path at each of its points - $\gamma'(s)$ in some parametrization - and uses complex multiplication to do the weighing.

It's well-defined because the differential $ \gamma'(s) ds $ can be understood to be invariant under reparametrization; "speeding up" the tracing out of the curve at a point by say a factor of $ k $ will amplify $ \gamma'(s)$ by $ k $ while dilating $ ds $ by the same and hence cancelling out. If one uses the natural parametrization (so $ |\gamma'| = 1 $), one can see that the path doesn't really "weigh" the function $ f$ by changing its magnitude but rather its phase, to what extend depending on the direction of the curve.

Complex analysis has bore a number of useful theorems for these integrals. For one, the fundamental theorem of calculus still applies: if the path $ \gamma $ goes from $ a $ to $ b $ in the complex plane, then $ \int_\gamma f'(z) dz = f(b) - f(a) $. This implies that any contour integral over a closed path vanishes, with the exception being when there is a pole inside. There's lots more to learn about this branch of mathematics but I'll leave that at that.

Answer 2

The question is a bit vague, so there are many things I can say. I think joriki has given a basic background, but I'd also recommend that you look up Jensen's Formula.

I think Jensen will show you something "magical", and this is sort of related to what you ask about a contour being related to a number. Let me first tell you what Jensen's Formula is:

Suppose that $\Omega$ is an open set that contains the closure of a disc $D_R$ and suppose that $f$ is holomorphic (complex-differentiable) in $\Omega$, $f(0)\neq 0$, and $f$ vanishes nowhere on the circle $C_R$ (the boundary of $D_R$). If $z_1 , z_2 , ..., z_N$ are the zeroes of $f$ inside the disc (count the multiplicities), then

$$\log \mid f(0) \mid=\sum_1 ^N log( |z_k| / R)+ 1/(2\pi) \int_C \log|f(z)|dz .$$

So, here "magically" a contour integration of a function (assume that it has no zeroes in the disc!) on the very right represents a number that somehow equals $\log|f(0)|$.

By the nature of your question, I know you haven't seen complex analysis before, so maybe some of the notions I mentioned don't make sense to you, but try to take in as much as you can! I am mentioning Jensen's Formula to you, because I think it sort of feeds your need for intuition when you ask how a contour represents a number (which is a very very broad question, so next time try to be mor specific!)