Timeline for What is the appropriate setting for Cauchy's Integral Formula?
Current License: CC BY-SA 3.0
11 events
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| May 15, 2017 at 15:30 | history | edited | Greg Zitelli | CC BY-SA 3.0 |
deleted 764 characters in body
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| Jan 12, 2017 at 2:07 | comment | added | Luke Burns | I'd like to add that a generalized integral formula is available in mixed signature Clifford algebras as well and so the result is not uniquely related to elliptic operators. See p261 Eqn (4.10) of the text "Clifford Algebra to Geometric Calculus." | |
| Apr 14, 2014 at 21:27 | comment | added | thomashennecke | The text "Generalized analytic functions" by Ilja Nestorowitsch Vekua, Pergamon press, 1962 is certainly quite old but maybe helpful?! | |
| Jul 16, 2012 at 20:55 | comment | added | Greg Zitelli | Igor, do you have any texts to recommend for the subject of general elliptic operators. | |
| Jun 30, 2012 at 9:10 | comment | added | Igor Khavkine | The Cauchy-Riemann equations are an elliptic system. Their solutions have properties that are special cases of results for somewhat more general elliptic systems: analyticity -- elliptic regularity, analytic continuation -- unique continuation, Cauchy integral formula -- Green function identity for Dirichlet boundary value property. | |
| Jun 30, 2012 at 2:02 | history | edited | Greg Zitelli | CC BY-SA 3.0 |
Appended more information
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| Jun 28, 2012 at 18:27 | comment | added | George Lowther | Just an idea - but you get similar integral identities for diffusions, where $f$ is in the kernel of the generator. This is just by writing $f(x)$ as the expected value of $f(X)$ at the first time at which $X$ hits the boundary, where $X$ is the diffusion started from $x$. On the appropriate Clifford algebra, the generator splits into a product of first order differential operators. So, it is enough for $f$ to be in the kernel of such a linear term in order for it to also be in the kernel of the generator. | |
| Jun 28, 2012 at 16:57 | history | edited | Greg Zitelli | CC BY-SA 3.0 |
Clarified what monogenic means; deleted 3 characters in body
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| Jun 28, 2012 at 16:56 | comment | added | Greg Zitelli | I phrased it poorly. When I say that f is left monogenic such that the sum is zero, what I really meant is that left monogenic functions are functions such that the sum of the e_j partial_j is always 0, which the function f is assumed to be (sim. right monogenic and g). | |
| Jun 28, 2012 at 15:44 | comment | added | JHM | what is 'monogenic'? | |
| Jun 27, 2012 at 20:45 | history | asked | Greg Zitelli | CC BY-SA 3.0 |