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Unformatted text preview: tic at z0 if it is differentiable at z0 and also in some neighborhood of z0.
If f is differentiable at every complex number, it is called entire.
Consequences Since the usual rules for differentiation (power, product, quotient, chain
rule) all follow formally from the same definition as that above, we can deduce that the
same rules hold for complex differentiation.
Geometric Interpretation of f'(z)†
Question What does f'(z) look like geometrically?
Answer We describe the magnitude and argument separately. First look at the magnitude
of f'(zo). For z near z0,
Ôf(z) - f(z0)Ô |f(z) - f(z0)|
|f'(z0)| ‡ Ô
Ô z - z0 Ô
|z - z0|
In other words, the magnitude of f '(z0) gives us an expansion factor; The distance
between points is expanded by a factor of |f'(z0)| near z0.
Now look at the direction (argument) of f'(zo): [Note that this only makes sense if
f'(z0) ≠ 0 -- otherwise the argument is not well defined.]
f(z) - f(z0)
z - z0
Therefore, the argument of f'(z0) is Arg[f(z) - f(z0)] - Arg[z -...
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