# derivative and differentiability of complex function

Let $f(z)$ be given uniquely in a neighborhood of the point $z$ in $\mathbb{C}$.  If the difference quotient

 $\frac{\Delta f}{\Delta z}\;=\;\frac{f(z+\Delta z)-f(z)}{\Delta z}$

tends to a finite limit $A$ as $\Delta z\to 0$, then $A$ is the of $f$ at the point $z$ and is denoted by

 $\displaystyle f^{\prime}(z)=A=\lim_{\Delta z\to 0}\frac{\Delta f}{\Delta z}.$ (1)

Thus the difference$\lambda=\frac{\Delta f}{\Delta z}-A$  tends to zero simultaneously with $\Delta z$, and $\Delta f$ has the expansion

 $\Delta f=A\Delta z+\lambda\Delta z.$

If we denote  $|\Delta z|\;=:\;\varrho$,  we have

 $\lambda\Delta z=\frac{\lambda\Delta z}{\varrho}\cdot\varrho\;=\;\langle\varrho\rangle\varrho$

where $\langle\varrho\rangle$ means a complex number vanishing when $|\Delta z|=\varrho\to 0$.  Consequently, (1) implies

 $\displaystyle\Delta f\;=\;A\Delta z+\langle\varrho\rangle\varrho$ (2)

in which  $A=f^{\prime}(z)$  and  $\varrho=|\Delta z|$.  It’s easily seen that the conditions (1) and (2) are equivalent.  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.

## References

• 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 DerivativeAndDifferentiabilityOfComplexFunction 2014-02-23 18:20:58 2014-02-23 18:20:58 pahio (2872) pahio (2872) 3 pahio (2872) Definition msc 30A99