Complex derivative

This applet shows how the initially rectangular grid would be mapped at the given location by the selected function (drag to move the grid o another position). Also, the two partial derivatives df/dx and df/dy are displayed using the image point f(z) as the origin. To satisfy Cauchy-Riemann requirement, the derivative bars must be same length, prependicular and magenta must be right from the green when looking away from the cursor

The two partial derivatives of the complex function can be approximately calculated using Newton's approximation law:

where represent small a small enough change in the x and y directions. The first derivative is along the real axis and other is along imaginary axis.

Cauchy-Riemann equation

In order for a function to be complex differentiable, both partial derivatives must satisfy the Cauchy-Riemann equation

In polar system, division by i rotates 90 degrees counter clockwise preserving r. In other words, the must be a 90 degree clockwise rotation from the . In the applet, these partial derivatives are shown by magenta (x) and green (y) lines that must have the same length and stay at the right angle to satisfy the equation.

The function may be are always differentiable, never differentiable or this may depend on if some conditions are satisfied. Differentiable (also called conformal) functions are "better behaved": for instance, they preserve orientation and angles.

When a function is differentiable, its complex derivative is

Notes on functions

Notes on specific functions:

  • f(z) = z, the simple identity map.
  • f(z) = (1+i)z - i is an example of linear mapping, when straight lines map to straight lines. The derivatives line up perfectly with the grids (for non linear mapping, they would not).
  • f(z) = z^2/4 is a non linear mapping as the most of the grid lines map to curves. However this function is always differentiable: the two partial derivatives stay at the same length and the right angle, satisfying the Cauchy-Riemann equation. Unlike the previous two examples, the derivatives vary with z.
  • f(z) = 5/z is dilated inversion map. Despite turning the grid "inside out", this function is also differentiable (except at the origin) and therefore orientation preserving. The derivatives get very large near the origin (why)?
  • f(z) = x + y + iy is defined in terms of the real and imaginary parts of z (x and y). Functions defined in terms of components tend to be non-differentiable, whereas functions defined in terms of z and the usual algebraic operations tend to be differentiable. While this is a linear map, it is not differentiable: the derivatives do not stay at the right angle and the Cauchy-Riemann test fails.
  • f(z) = x + y^2 + iy is a non-linear variant of the previous map. This function in general is not differentiable, however it is differentiable for points at the real (x) axis (solve Cauchy-Riemann equation to check).
  • f(z) = exp(z), the exponential map that, while drastically altering the grid, is everywhere differentiable. The real part defines shrinking/growing while the imaginary part defines the rotation.
  • f(z) = zbar is the example of a non-differentiable function that preserves angles but not orientation. df/dx (magenta in the applet) is 90 degrees counter-clockwise from df/dy and not clockwise as it would be for identity map. This is enough to make the function not differentiable.