This post explores how complex-valued functions can be constructed from real-valued components, using the relationship between real and imaginary parts governed by the Cauchy-Riemann equations. It demonstrates practical techniques for building analytic functions from real harmonic functions, bridging real analysis and complex analysis.
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).