Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

7: Complex Derivatives

( \newcommand{\kernel}{\mathrm{null}\,}\)

We have studied functions that take real inputs and give complex outputs (e.g., complex solutions to the damped harmonic oscillator, which are complex functions of time). For such functions, the derivative with respect to its real input is much like the derivative of a real function of real inputs. It is equivalent to taking the derivatives of the real and imaginary parts, separately: dψdx=dRe(ψ)dx+idIm(ψ)dx.

Now consider the more complicated case of a function of a complex variable: f(z)C,wherezC.
At one level, we could just treat this as a function of two independent real inputs: f(x,y), where z=x+iy. However, in doing so we would be disregarding the mathematical structure of the complex input—the fact that z is not merely a collection of two real numbers, but a complex number that can participate in algebraic operations. This structure has important implications for the differential calculus of complex functions.


This page titled 7: Complex Derivatives is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?