The complex derivative

This applet displays a complex map w=f(z) as in Applet 2, but with more features. Firstly, a grid is displayed on the domain, and the image of the grid on the range. Clicking the mouse on the domain will redraw the grid centered at the current location. Secondly, the two partial derivatives df/dx and df/dy are displayed in green and cyan respectively; using the image point f(z) as the origin rather than the standard origin. (You can change the color scheme, of course). If you move the mouse to the right, the image will move in the direction of the x derivative; if you move the mouse upward, the image will move in the direction of the y derivative.

Note that the green and cyan lines are tangent to the grid. This is in accordance with Newton's approximation law

 f(z + dx)   ~ f(z) + df/dx  dx

 f(z + i dy) ~ f(z) + df/dy  dy

In order for a function w=f(z) to be complex differentiable, the two partial derivatives must satisfy the Cauchy-Riemann equation df/dx = (1/i) df/dy. In other words, the green line must be a 90 degree clockwise rotation of the cyan line. Some of the functions listed here are always differentiable; some are never differentiable; and one is differentiable sometimes and non-differentiable other times.

Differentiable functions are much better behaved than non-differentiable functions; for instance, they preserve orientation and angles. Another name for this is conformal.

When a function is differentiable, its complex derivative df/dz is given by the equation df/dz = df/dx = (1/i) df/dy. In other words, the complex derivative co-incides with the green line, and with the 90-degree clockwise rotation of the cyan line.

Notes on specific functions:

• f(z) = z: Once again, the boring old identity map.

• f(z) = (1+i)z - i: Straight lines map to straight lines, which is why these maps are sometimes called "linear". Notice that the derivatives line up perfectly with the grids.

• f(z) = z^2/4: Here's a map which isn't linear; most of the grid lines map to curves. Nevertheless, the function is always differentiable; notice how the two partial derivatives are 90 degree rotations of each other. Unlike the previous two examples, the derivatives of f are not constant, but vary with z. In fact, there's a simple relationship between the derivative df/dz and the original complex number z; what is it? It should not be surprising.

• f(z) = 5/z: A dilated inversion map. Despite turning the grid "inside out", the map is differentiable and therefore orientation preserving. (except at the origin, where it is not even well-defined). The derivative can get very large near the origin. Why?

• f(z) = zbar: Here is a non-differentiable function: it preserves angles but not orientation. In what way does the Cauchy-Riemann equations fail?

• f(z) = x + y + iy: Here is a function defined in terms of the real and imaginary parts of z. As a rule of thumb, 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. Is this function differentiable anywhere? Does it preserve orientation? angles?

• f(z) = x + y^2 + iy: A non-linear variant of the previous map. This function is differentiable at some places, but not at others. Can you figure out where it is differentiable? You can solve the Cauchy-Riemann equations to obtain an answer.

• f(z) = exp(z). What happens to the horizontal grid lines under this map? The vertical grid lines? Can you explain this?

  The exponential map is everywhere differentiable. There is a very obvious relationship between the derivative df/dz and the original function f(z). What is it? It should not be too surprising.

  What is the relationship between df/dy and f? When you move z up and down, what happens to df/dy and f(z)?

Next applet: Complex integration

Previous applet: Multiple-valued functions

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