Complex Differentiable
المؤلف:
Shilov, G. E
المصدر:
Elementary Real and Complex Analysis. New York: Dover
الجزء والصفحة:
p. 379
18-10-2018
868
Complex Differentiable
Let 
and 
on some region 
containing the point 
. If 
satisfies the Cauchy-Riemann equations and has continuous first partial derivatives in the neighborhood of 
, then 
exists and is given by
and the function is said to be complex differentiable (or, equivalently, analytic or holomorphic).
A function 
can be thought of as a map from the plane to the plane, 
. Then 
is complex differentiable iff its Jacobian is of the form
at every point. That is, its derivative is given by the multiplication of a complex number 
. For instance, the function 
, where 
is the complex conjugate, is not complex differentiable.
REFERENCES:
Shilov, G. E. Elementary Real and Complex Analysis. New York: Dover, p. 379, 1996.
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