The article shows how to build complex analytic functions from their real parts using the Schwarz reflection principle and analytic continuation, recovering a function from real-axis data up to an additive imaginary constant.
The article explains how complex elementary functions, such as the sine and cosine of a complex number, can be computed using only real functions of real variables. While some functions require more complicated equations than the basic examples, this approach can be extended to all elementary functions.
John D. Cook discusses notes based on Henry Baker's approach to implementing complex-variable functions using real-variable functions, decomposing f(x+iy) into real and imaginary parts u(x,y) and v(x,y).