To find this ratio, Seidl and Klose  converted the Cauchy-Riemann equations into a set of two Laplace equations and proposed an iterative procedure with a finite difference solution.
In this paper, by applying the rectangular properties of the transformed region and the Cauchy-Riemann equations, the derivatives and the Jacobian determinants of the transformation can be evaluated on the boundaries.
Based on the Cauchy-Riemann equations, the forward conformal transformation, from the physical domain (x-y coordinates) to the computational domain ([xi]-[eta] coordinates), can be reduced to a problem governed by two Laplace equations:
Leonhard Euler's identity, the Pythagorean identity and the Cauchy-Riemann equations
were the formula most consistently rated as beautiful.
The formulae most consistently rated as beautiful (both before and during the scans) were Leonhard Euler's identity, the Pythagorean identity and the Cauchy-Riemann equations
. Leonhard Euler's identity links five fundamental mathematical constants with three basic arithmetic operations each occurring once and the beauty of this equation has been likened to that of the soliloquy in Hamlet.
This is well defined using the Cauchy-Riemann equations
for the coordinate interchanges.
In other words, the Cauchy-Riemann equations
need not to be satisfied, so the functions need not to be analytic (Duren, 2004).
He starts with Cauchy-Riemann equations
in the introduction, then proceeds to power series, results on holomorphic functions, logarithms, winding numbers, Couchy's theorem, counting zeros and the open mapping theorem, Eulers formula for sin(z), inverses of holomorphic maps, conformal mappings, normal families and the Riemann mapping theorem, harmonic functions, simply connected open sets, Runge's theorem and the Mittag-Leffler theorem, the Weierstrass factorization theorem, Caratheodory's theorem, analytic continuation, orientation, the modular function, and the promised Picard theorems.
the generalized Cauchy-Riemann equations
contain 2-spinor and C-gauge structures, and their integrability conditions take the form of Maxwell and Yang-Mills equations.
For instance, the Cauchy-Riemann equations
, which specify the regularity conditions for a complex-valued function to be analytic (expressible as a power series), generalize to the Lanczos equations in Minkowski spacetime, and then generalize further to the Nijenhuis tensor equations for holomorphic functions in n-dimensional space.
Then one looks for other type of decompositions, where the Cauchy-Riemann equations nevertheless express the vanishing of some particular term.
As anyone who has worked in the field knows, this quality is like a divide in the quaternionic world, separating the canonical objects from the non canonical ones: of course, the Cauchy-Riemann equations also belong to the first category.