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Subject: gnuplot / spherical_harmonics / spharm (2)
Date: Mon, 21 Aug 2006 20:44:18 +0900
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      <DIV align=3Dcenter>
      <H3>- not so Frequently Asked Questions - </H3></DIV>
      <DIV class=3Dupdate>update 2004/11/16 </DIV></TD></TR>
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    <TD id=3Dcontent>
      <H1><A =
name=3Dtop>=8E=ED=81X=82=CC=8BO=93=B9=8Ap=89^=93=AE=97=CA=82=C6=8E=A5=8BC=
=97=CA=8Eq=90=94 </A></H1>
      <H2><A name=3DL1>l=3D1,m=3D-1,+1 </A></H2>
      =
<P>=8B=85=96=CA=92=B2=98a=8A=D6=90=94Y[lm]=82=CCl,m=82=CD=81C=97=CA=8Eq=97=
=CD=8Aw=82=CC=8BO=93=B9=8Ap=89^=93=AE=97=CA=82=C6=8E=A5=8BC=97=CA=8Eq=90=94=
=82=C5=82=B7=81D=82=B1=82=B1=82=C5=82=CD=81C=8F=AC=82=B3=82=A2l=82=C9=91=CE=
=82=B7=82=E9=83O=83=89=83t=82=F0=8A=F4=82=C2=82=A9=95\=8E=A6=82=B5=82=C4=82=
=DD=82=DC=82=B7=81D=82=E2=82=E8=95=FB=82=CD=81C=91O=8D=80=82=C6=93=AF=97l=
=81C=94}=89=EE=95=CF=90=94=82=F0=97p=82=A2=82=C4x,y,z=82=F0=95\=82=B7=95=FB=
=96@=82=C5=82=B7=81D</P>
      =
<P>m=82=AA=8A=EF=90=94=82=CC=8F=EA=8D=87=82=CD=8B=85=96=CA=92=B2=98a=8A=D6=
=90=94=82=CDexp(-im=83=D3)=82=CC=8D=80=82=CC=88=D7=82=C9=8B=95=95=94=82=F0=
=8E=9D=82=BF=82=DC=82=B7=81D =
m=3D-1,1=82=CC=82=CC=82=B1=82=CC=8A=D6=90=94=82=CD=8E=9F=82=CC=82=E6=82=A4=
=82=C9=82=C8=82=E8=82=DC=82=B7=81D</P><PRE>   Y[1,-1](t,p) =3D  =
sqrt(3/8pi) sin(t) exp(-i p)
   Y[1, 1](t,p) =3D -sqrt(3/8pi) sin(t) exp( i p)
</PRE>
      =
<P>|Y|^2=82=CDY=82=C6=82=BB=82=CC=95=A1=91f=8B=A4=96=F0=82=F0=82=A9=82=AF=
=82=EA=82=CE=82=E6=82=AD=81C=8F=E3=82=CC2=82=C2=82=CC=8E=AE=82=CD=82=A2=82=
=C3=82=EA=82=E0=93=AF=82=B6Y (=83=C6)=3D3/8=83=CE sin^2=83=C6=20
      =
=82=C6=82=C8=82=E8=82=DC=82=B7=81D=82=C8=82=A8=81C=8B=85=96=CA=92=B2=98a=8A=
=D6=90=94=82=CC=95=A1=91f=8B=A4=96=F0=82=CD=8A=C8=92P=82=C9=8B=81=82=DF=82=
=E9=82=B1=82=C6=82=AA=82=C5=82=AB=81C<TT> (Y[l,m])^* =3D (-1)^m Y[l,-m]=20
      =
</TT>=82=C5=97^=82=A6=82=E7=82=EA=82=DC=82=B7=81D=82=C2=82=DC=82=E8=81Cm=82=
=CC=95=84=8D=86=82=F0=95=CF=82=A6=81C=82=B3=82=E7=82=C9m=82=AA=8A=EF=90=94=
=82=CC=8F=EA=8D=87=82=CD=91S=91=CC=82=CC=95=84=8D=86=82=F0=95=CF=82=A6=82=
=DC=82=B7=81D</P><PRE class=3Dsample>gnuplot&gt; set parametric

        dummy variable is t for curves, u/v for surfaces
gnuplot&gt; set angle degree
gnuplot&gt; set urange [0:360]
gnuplot&gt; set vrange [-90:90]
gnuplot&gt; set isosample 36,18
gnuplot&gt; set ticslevel 0
gnuplot&gt; set size 0.65,1.0
gnuplot&gt; a=3D3.0/(8*pi)
gnuplot&gt; fx(u,v)=3Dcos(u)*cos(v)
gnuplot&gt; fy(u,v)=3Dsin(u)*cos(v)
gnuplot&gt; fz(v)=3Dsin(v)
gnuplot&gt; g(v)=3Dcos(v)*cos(v)
gnuplot&gt; splot a*g(v)*fx(u,v),a*g(v)*fy(u,v),a*g(v)*fz(v)
</PRE><IMG height=3D300 alt=3Dsph11=20
      =
src=3D"http://t16web.lanl.gov/Kawano/gnuplot/spherical_harmonics/sph11.pn=
g"=20
      width=3D400 align=3Dleft>=20
      =
<P>=8D=B6=82=AAY[11]=82=F0=8E=A9=8F=E6=82=B5=82=BD=83O=83=89=83t=82=C5=81=
Cz=8E=B2=82=F0=8E=E5=8E=B2=82=C9=82=B5=82=BD=83g=81[=83=89=83X=82=C6=82=C8=
=82=E8=82=DC=82=B7=81D=92A=82=B5=20
      =
=93=E0=91=A4=82=C9=8C=8A=82=CD=82=A0=82=E8=82=DC=82=B9=82=F1=81D=8A=D6=90=
=94=82=CDsin(=83=C6)=82=F0=8A=DC=82=DD=82=DC=82=B7=82=AA=81C=8Ap=93x=83=C6=
=82=F0<I>v</I>=82=C5 =
=95\=82=B7=82=C6cos(v)=82=C9=82=C8=82=E9=82=CC=82=C5=81C<TT>=20
      g(v)=3Dcos(v)*cos(v)</TT>=82=C6=8A=D6=90=94=92=E8=8B`=82=B5=82=C4 =
=82=A2=82=DC=82=B7=81D</P>
      <DIV class=3Dtop><A=20
      =
href=3D"http://t16web.lanl.gov/Kawano/gnuplot/spherical_harmonics/spharm2=
.html#top"><IMG=20
      alt=3Dup=20
      =
src=3D"http://t16web.lanl.gov/Kawano/gnuplot/image/mark-up.png"></A></DIV=
>
      <H2><A name=3DL2>l=3D2,m=3D-2,-1,0,+1,+2 </A></H2>
      =
<P>l=3D2=82=CC=82=C6=82=AB=82=CD=81C=8E=A5=8BC=97=CA=8Eq=90=94m=82=AA =
-2=81`2 =
=82=DC=82=C5=82=CC5=82=C2=82=CC=8A=D6=90=94=82=AA=82=A0=82=E8=82=DC=82=B7=
=81D=82=BB=82=EA=82=BC=82=EA=82=CC=8A=D6=90=94=82=CD=88=C8=89=BA=82=C5=97=
^=82=A6=82=E7=82=EA=82=DC=82=B7=81D</P><PRE>   Y[2,-2](t,p) =3D  =
sqrt(15/32pi) sin(t)sin(t) exp(-2i p)
   Y[2,-1](t,p) =3D  sqrt(15/8pi) sin(t)cos(t) exp(-i p)
   Y[2, 0](t,p) =3D  sqrt(5/16pi) (3cos(t)cos(t)-1)
   Y[2, 1](t,p) =3D -sqrt(15/8pi) sin(t)cos(t) exp( i p)
   Y[2, 2](t,p) =3D  sqrt(15/32pi) sin(t)sin(t) exp( 2i p)
</PRE>
      =
<P>=8E=A9=8F=E6=82=C9=82=B5=82=C4=95\=8E=A6=82=B7=82=E9=82=C6|m|=82=AA=93=
=AF=82=B6=82=E0=82=CC=82=CD=93=AF=82=B6=83O=83=89=83t=82=C9=82=C8=82=E8=82=
=DC=82=B7=82=CC=82=C5=81Cm=3D0,1,2=82=CC =
3=82=C2=82=CC=8F=EA=8D=87=82=C9=82=C2=82=A2=82=C4=95\=8E=A6=82=B5=82=DC=82=
=B7=81D</P>
      <P><B>m=3D0 </B></P><PRE class=3Dsample>gnuplot&gt; a=3D =
5.0/(16*pi)
gnuplot&gt; g(v)=3D (3*sin(v)*sin(v)-1)**2
gnuplot&gt; splot a*g(v)*fx(u,v),a*g(v)*fy(u,v),a*g(v)*fz(v)
</PRE>
      <DIV align=3Dcenter><IMG height=3D300 alt=3Dsph20=20
      =
src=3D"http://t16web.lanl.gov/Kawano/gnuplot/spherical_harmonics/sph20.pn=
g"=20
      width=3D400></DIV><BR>
      <P><B>m=3D1 </B></P><PRE class=3Dsample>gnuplot&gt; a=3D15.0/( =
8*pi)
gnuplot&gt; g(v)=3D (sin(v)*cos(v))**2
gnuplot&gt; splot a*g(v)*fx(u,v),a*g(v)*fy(u,v),a*g(v)*fz(v)
</PRE>
      <DIV align=3Dcenter><IMG height=3D300 alt=3Dsph21=20
      =
src=3D"http://t16web.lanl.gov/Kawano/gnuplot/spherical_harmonics/sph21.pn=
g"=20
      width=3D400></DIV><BR>
      <P><B>m=3D2 </B></P><PRE class=3Dsample>gnuplot&gt; =
a=3D15.0/(32*pi)
gnuplot&gt; g(v)=3D (cos(v)*cos(v))**2
gnuplot&gt; splot a*g(v)*fx(u,v),a*g(v)*fy(u,v),a*g(v)*fz(v)
</PRE>
      <DIV align=3Dcenter><IMG height=3D300 alt=3Dsph22=20
      =
src=3D"http://t16web.lanl.gov/Kawano/gnuplot/spherical_harmonics/sph22.pn=
g"=20
      width=3D400></DIV><BR>
      <DIV class=3Dtop><A=20
      =
href=3D"http://t16web.lanl.gov/Kawano/gnuplot/spherical_harmonics/spharm2=
.html#top"><IMG=20
      alt=3Dup=20
      =
src=3D"http://t16web.lanl.gov/Kawano/gnuplot/image/mark-up.png"></A></DIV=
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