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 dywhere dx, dy represent small increments in the x and y directions.
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:
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
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