proof to Cauchy-Riemann equations (polar coordinates)
If f(z) is differentialble at z0 then the following limit
f′(z0) | = | lim |
will remain the same approaching from any direction. First we fix as then we take the limit along the ray where the argument is equal to . Then
Similarly, if we take the limit along the circle with fixed equals . Then
Note: We use l’Hôpital’s rule to obtain the following result used above .
Now, since the limit is the same along the circle and the ray then they are equal:
which implies that
QED
Title | proof to Cauchy-Riemann equations![]() ![]() |
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Canonical name | ProofToCauchyRiemannEquationspolarCoordinates |
Date of creation | 2013-03-22 14:06:13 |
Last modified on | 2013-03-22 14:06:13 |
Owner | Daume (40) |
Last modified by | Daume (40) |
Numerical id | 5 |
Author | Daume (40) |
Entry type | Proof |
Classification | msc 30E99 |