derivative and differentiability of complex function

Let f(z) be given uniquely in a neighborhood of the point z in .  If the difference quotient


tends to a finite limit A as Δz0, then A is the derivativeMathworldPlanetmath of f at the point z and is denoted by

f(z)=A=limΔz0ΔfΔz. (1)

Thus the differencePlanetmathPlanetmathλ=ΔfΔz-A  tends to zero simultaneously with Δz, and Δf has the expansion


If we denote  |Δz|=:ϱ,  we have


where ϱ means a complex numberPlanetmathPlanetmath vanishing when |Δz|=ϱ0.  Consequently, (1) implies

Δf=AΔz+ϱϱ (2)

in which  A=f(z)  and  ϱ=|Δz|.  It’s easily seen that the conditions (1) and (2) are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath.  The latter expresses the differentiability of f at z.  By it one can sayt that the increment of f is “locally proportional” to the increment of z.  Cf. the consideration of differential of real functions.


  • 1 E. Lindelöf: Johdatus funktioteoriaan (‘Introduction to function theory’).  Mercatorin kirjapaino, Helsinki (1936).
  • 2 R. Nevanlinna & V. Paatero: Funktioteoria.  Kustannusosakeyhtiö Otava, Helsinki (1963).

Title derivative and differentiability of complex function
Canonical name DerivativeAndDifferentiabilityOfComplexFunction
Date of creation 2014-02-23 18:20:58
Last modified on 2014-02-23 18:20:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 3
Author pahio (2872)
Entry type Definition
Classification msc 30A99