If C is a curve in the complex plane joining the points z_{0} and z_{1}, the line integral of a function f(z)=u+i v along C is defined by the equation
The real and imaginary parts of (4.1) are thus ordinary real line integrals in the plane, of the type considered in red{calculus}. We see that the conditions that the two integrands be exact differentials are precisely the requirement that the Cauchy-Riemann equations \dfrac{\partial u}{\partial x}=\dfrac{\partial v}{\partial y} and \dfrac{\partial u}{\partial y}=-\dfrac{\partial v}{\partial x} (3.8a,b) be satisfied. The results of the ordinary real integrals and Chapter 3 then show that:
The line integral \displaystyle \int_{C} f(z) \,d z is independent of the path C joining the end points z_{0} and z_{1} if C can be enclosed in a simply connected region R inside which f(z) is analytic.
In such a case the curve need not be prescribed and we may indicate the integral by the notation \int_{z_{0}}^{z_{1}} f(z) \,d z. It follows also that here f(z) \,d z is the exact differential of a function F(z),
and that the integral can then be integrated in the usual way, according to the formula
where F(z) is a function whose derivative is f(z).
By considering the case where the end points z_{0} and z_{1} coincide, we arrive at the conclusion that:
