ALPHONSE KABR, Les Guépes, 1849
Let ζ and z be two complex variables related by
where f(ζ) is an analytic function. Then if ζ moves along a curve in the plane of ξ, η, z will move along a curve in the plane of x, y. Every value of ζ that f(ζ) is defined for will identify a point in the x, y plane, and conversely if the inverse function exists for a value of z a value of ζ will be identified. Thus (1) can be regarded as a transformation, enabling us to map at least a part of the ξ, η plane on the x, y plane and conversely.