Holomorphic Function Explorer

In this webapp you can play with functions from ℂ to ℂ and their derivatives. Draw with your mouse on the left side and see your transformed drawing on the right side!

You can input a complex function with one variable 𝑧, all the regular operators such as + - * / ^ , re() , im() , sqrt() and abs() are suppored. In general, we support anything that math.js can parse.

Make sure to adjust the zoom level on each side for the perfect experience.

As you mouse over the graph, the complex derivative will be displayed as a 2D linear transformation with red and green arrows for the basis. Check for youself that:

If you draw around the derivative the drawing will be transformed pretty linearly. The derivative preserves angles! Cauchy-Riemann FTW.

Functions without derivatives

If you input a function without complex derivatives, such as e^z+re(z) , we will "approximate" the derivative by treating the function as ℝ²🠂ℝ². Then you can see for yourself how the Cauchy-Riemann equations break down as the derivative no longer preserves angles.

Cool functions to try

e^z

e^z+re(z)

(z+2)^2(z-1-2i)(z+1)

sqrt(z)

log(z)*re(z)

z*(abs(re(z))+abs(im(z)))

Play!

If you find any interesting functions please let me know.