Hartogs’s theorem on separate analyticity
This is a of an analogue of Goursat’s theorem for several complex variables. That is if we just consider that a function is holomorphic in each variable separately, then it will turn out to be continuously differentiable. Thus we can in fact define holomorphic functions of several complex variables to be just functions holomorphic in each variable separately.
Note that there is no analogue of this theorem for real variables. If we assume that a function is differentiable (or even analytic) in each variable separately, it is not true that will necessarily be continuous. The standard example in is given by , then this function has well defined partial derivatives in and at 0, but it is not continuous at 0 (try approaching 0 along the line or ).
Even if we assume the function to be smooth (), there is no analogue for real variables. Consider where we define This function is smooth, real analytic in each variable separately, but it fails to be real analytic at the origin.
- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
|Title||Hartogs’s theorem on separate analyticity|
|Date of creation||2013-03-22 14:29:24|
|Last modified on||2013-03-22 14:29:24|
|Last modified by||jirka (4157)|