 In real variables, we introduce the definite integral by partitioning our interval into n subintervals with delta xi, evaluating our function at some point within each subinterval, summing the products, and then we took the limit as the greatest width went to zero. But if we try this for complex functions we run into a problem, what is the interval between two points? So we might consider a path from z0 to z1, and to begin with let's require that our path be continuous, so something with a break in it would not be allowed, non-self-intersecting, so something that crossed itself would not be allowed, and piecewise smooth, so the occasional corner is all right. And if we meet all these requirements we say that c is a simple path. This allows us to parameterize z as some function of t over some interval, where the derivative exists everywhere except perhaps at some isolated points. Then we get the line integral, and remember the more important something is, the more words we have to describe it. So this line integral is also called a contour or a path integral. So for example let's evaluate this contour integral where c is the straight line from z equals 1 to z equals i. So we need to parameterize the path. Travelling from z equals 1 to z equals i along a straight line path is equivalent to traveling along the straight line from 1, 0 to 0, 1. And we can parameterize this path as, so if we treat i as a constant, which actually it is, then from z equals x of t plus i y of t we can find z and dz, and then substituting them into our integral gives us. Now because c is the path from z equals 1 to z equals i, we could claim that the integral from 1 to i is negative 1. Or could we? The problem is the limits of the complex integral don't restrict our paths. So what if we took a different path between the two points? So let's consider this as the arc of the unit circle centered at the origin from z equals 1 to z equals i. So we can parameterize that arc as, and again letting z be cosine t plus i sine t, then dz will be, and making our substitutions gives us. We could find the contour integral for other paths, but we eventually find that our integral is negative 1 for any path between z equals 1 and z equals i. And this leads to a remarkable result. The value of an integral depends only on the value of a function at its endpoints. We'll see some implications of this next.