Poisson Kernel
المؤلف:
Gradshteyn, I. S. and Ryzhik, I. M
المصدر:
Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press
الجزء والصفحة:
...
27-12-2018
1779
Poisson Kernel
The integral kernel in the Poisson integral, given by
 |
(1)
|
for the open unit disk 
. Writing 
and taking 
gives
(Krantz 1999, p. 93).
In three dimensions,
 |
(7)
|
where 
and
![cosgamma=y·[Rcosthetasinphi; Rsinthetasinphi; Rcosphi].](http://mathworld.wolfram.com/images/equations/PoissonKernel/NumberedEquation3.gif) |
(8)
|
The Poisson kernel for the 
-ball is
 |
(9)
|
where 
is the outward normal derivative at point 
on a unit 
-sphere and
 |
(10)
|
Let 
be harmonic on a neighborhood of the closed unit disk 
, then the reproducing property of the Poisson kernel states that for 
,
 |
(11)
|
(Krantz 1999, p. 94).
REFERENCES:
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1090, 2000.
Krantz, S. G. "The Poisson Kernel." §7.3.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 93, 1999.
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