sexta-feira, 20 de maio de 2016

Função Harmónica

Harmonic Function

Any real function u(x,y) with continuous second partial derivatives which satisfies Laplace's equation,
del ^2u(x,y)=0,
(1)
is called a harmonic function. Harmonic functions are called potential functions in physics and engineering. Potential functions are extremely useful, for example, in electromagnetism, where they reduce the study of a 3-component vector field to a 1-component scalar function. A scalar harmonic function is called a scalar potential, and a vector harmonic function is called a vector potential.
To find a class of such functions in the plane, write the Laplace's equation in polar coordinates
u_(rr)+1/ru_r+1/(r^2)u_(thetatheta)=0,
(2)
and consider only radial solutions
u_(rr)+1/ru_r=0.
(3)
This is integrable by quadrature, so define v=du/dr,
(dv)/(dr)+1/rv=0
(4)
(dv)/v=-(dr)/r
(5)
ln(v/A)=-lnr
(6)
v/A=1/r
(7)
v=(du)/(dr)=A/r
(8)
du=A(dr)/r,
(9)
so the solution is
u=Alnr.
(10)
Ignoring the trivial additive and multiplicative constants, the general pure radial solution then becomes
u=ln[(x-a)^2+(y-b)^2]^(1/2)
(11)
=1/2ln[(x-a)^2+(y-b)^2].
(12)
Other solutions may be obtained by differentiation, such as
u=(x-a)/((x-a)^2+(y-b)^2)
(13)
v=(y-b)/((x-a)^2+(y-b)^2),
(14)
u=e^xsiny
(15)
v=e^xcosy,
(16)
and
tan^(-1)((y-b)/(x-a)).
(17)
Harmonic functions containing azimuthal dependence include
u=r^ncos(ntheta)
(18)
v=r^nsin(ntheta).
(19)
u(r,R,theta,phi)=(R^2-r^2)/(R^2-2rRcos(theta-phi)+r^2)
(20)
is another harmonic function.
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