Obtaining analytic functions and conjugate harmonic functions
d\J/ + ' ~~8x ̂ ~ '•y°' •Vo) dz (1) in the "variable" z. These formulas are readily obtained from the Cauchy-Riemann equations once it is observed that f(z) = <fi(z iy0, y0) + i\p(z />„, y0) (2) holds for all z in some neighborhood of z0. Actually, Laitone derived (2) only for y0 = 0, using the conjugate complex function /(z). An alternative (and perhaps more direct and illuminating) derivation can be given using Taylor series. For if 0(W + x0, y0) = ^anu\ \J/(u + x0, y0) = J^bnun (3) are the Taylor series expansions (in the first variable) of 0 and \p about x0, then 1 8n4> , x , 1 8ni , (*<" ̂ K = ^.~e^(*°' hold for all n. But dnj/dzn = 8n4>/8xn + id"\p/8xn, so that the Taylor series expansion of/ about z0 = x0 + iy0 gives