Namespaces
Variants
Actions

Difference between revisions of "Zonal harmonics"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(latex details)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
 +
 +
Out of 44 formulas, 44 were replaced by TEX code.-->
 +
 +
{{TEX|semi-auto}}{{TEX|done}}
 
''zonal harmonic polynomials''
 
''zonal harmonic polynomials''
  
 
Zonal harmic polynomials are spherical harmonic polynomials (cf. also [[Spherical harmonics|Spherical harmonics]]) that assume constant values on circles centred on an axis of symmetry. They characterize single-valued harmonic functions on simply-connected domains with rotational symmetry.
 
Zonal harmic polynomials are spherical harmonic polynomials (cf. also [[Spherical harmonics|Spherical harmonics]]) that assume constant values on circles centred on an axis of symmetry. They characterize single-valued harmonic functions on simply-connected domains with rotational symmetry.
  
To be specific, one introduces the [[Spherical coordinates|spherical coordinates]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z1301301.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z1301302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z1301303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z1301304.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z1301305.png" />. The zonal harmonics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z1301306.png" /> are the polynomial solutions of the [[Laplace equation|Laplace equation]]
+
To be specific, one introduces the [[Spherical coordinates|spherical coordinates]] $( r , \theta , \varphi )$ as $x _ { 1 } = r \operatorname { sin } \theta \operatorname { cos } \varphi$, $x _ { 2 } = r \operatorname { sin } \theta \operatorname{sin} \phi$, $x _ { 3 } = r \operatorname { cos } \theta$, where $( x _ { 1 } , x _ { 2 } , x _ { 3 } ) \in \mathbf{R} ^ { 3 }$. The zonal harmonics $H _ { n }$ are the polynomial solutions of the [[Laplace equation|Laplace equation]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z1301307.png" /></td> </tr></table>
+
\begin{equation*} \left[ \partial _ { r r } + \frac { 2 } { r } \partial _ { r } + \frac { 1 } { r ^ { 2 } } \partial _ { \theta \theta } + \frac { \operatorname { ctan } \theta } { r ^ { 2 } } \partial _ { \theta } + \frac { 1 } { r ^ { 2 } \operatorname { sin } ^ { 2 } \theta } \partial _ { \varphi \varphi } \right] H = 0 \end{equation*}
  
that are axially symmetric (i.e. independent of the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z1301308.png" />). They can be expressed in terms of [[Legendre polynomials|Legendre polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z1301309.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013010.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013012.png" />, and form a complete orthogonal set of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013014.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013015.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013016.png" /> vanish on cones that divide a sphere centred at the origin into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013017.png" /> zones, hence the name zonal harmonics. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013018.png" /> are sometimes referred to as solid zonal harmonics and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013019.png" /> as surface zonal harmonics.
+
that are axially symmetric (i.e. independent of the angle $\varphi$). They can be expressed in terms of [[Legendre polynomials|Legendre polynomials]] $P_n$ of degree $n$, as $H _ { n } ( r , \theta ) = r ^ { n } P _ { n } ( \operatorname { cos } \theta )$ for $n = 0,1 , \dots$, and form a complete orthogonal set of functions in $L ^ { 2 } [ D ]$, where $D$: $r \leq r_0$. The $H _ { n }$ vanish on cones that divide a sphere centred at the origin into $n$ zones, hence the name zonal harmonics. The $H _ { n }$ are sometimes referred to as solid zonal harmonics and the $P_n$ as surface zonal harmonics.
  
 
==Applications.==
 
==Applications.==
 
Two types of applications arise in classical [[Potential theory|potential theory]] (see [[#References|[a4]]], [[#References|[a6]]], [[#References|[a7]]]).
 
Two types of applications arise in classical [[Potential theory|potential theory]] (see [[#References|[a4]]], [[#References|[a6]]], [[#References|[a7]]]).
  
In the first, one determines the potential in a sphere from its boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013020.png" />. By specifying appropriate regularity conditions, the orthogonality of the Legendre polynomials is used to expand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013021.png" /> as the Fourier–Legendre series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013022.png" />. The potential in the sphere is recovered as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013023.png" />. The exterior boundary value problem is formulated by means of the [[Kelvin transformation|Kelvin transformation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013024.png" />. The potential between two concentric spheres is determined by combining solutions of the interior and the exterior problems.
+
In the first, one determines the potential in a sphere from its boundary values $H ( r _ { 0 } , \theta )$. By specifying appropriate regularity conditions, the orthogonality of the Legendre polynomials is used to expand $H ( r _ { 0 } , \theta )$ as the Fourier–Legendre series $\sum _ { n = 0 } ^ { \infty } a _ { n } n_{0} ^ { n } P _ { n } ( \operatorname { cos } \theta )$. The potential in the sphere is recovered as $H ( r , \theta )$. The exterior boundary value problem is formulated by means of the [[Kelvin transformation|Kelvin transformation]] $H ( r , \theta ) \rightarrow ( 1 / r ) H ( 1 / r ^ { 2 } , \theta )$. The potential between two concentric spheres is determined by combining solutions of the interior and the exterior problems.
  
In the second, one determines the potential at points in space from its values on a segment of the symmetry axis. The solution relies on the fact that along this axis the zonal harmonics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013026.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013028.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013030.png" /> is the radius of convergence of the Taylor series.
+
In the second, one determines the potential at points in space from its values on a segment of the symmetry axis. The solution relies on the fact that along this axis the zonal harmonics $H _ { n } ( r , 0 ) = r ^ { n }$, $n = 0,1 , \dots$. Thus, if $H ( r , 0 ) = \sum _ { n = 0 } ^ { \infty } a _ { n } H _ { n } ( r , 0 )$, then $H ( r , \theta ) = \sum _ { n = 0 } ^ { \infty } a _ { n } H _ { n } ( r , \theta )$ for $r < r_{0}$, where $r_0$ is the radius of convergence of the Taylor series.
  
 
==Relation with analytic functions.==
 
==Relation with analytic functions.==
There are many connections between the properties of the potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013031.png" /> and those of analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013032.png" /> of a complex variable (cf. also [[Analytic function|Analytic function]]; [[Harmonic function|Harmonic function]]). One such connection, related to the previous example, concerns singularities and uses the generating function for zonal harmonics to construct reciprocal integral transforms connecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013033.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013034.png" />. The following fact is immediate (see [[#References|[a3]]], [[#References|[a8]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013035.png" /> be a sequence of real constants for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013036.png" />. Consider the associated harmonic and analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013038.png" />, which are regular for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013039.png" />. Then the boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013040.png" /> is a singularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013041.png" /> if and only if the boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013042.png" /> is a singularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013043.png" />. Thus, the singularities of solutions of a singular partial differential equation are characterized in terms of those of associated analytic functions and vice versa.
+
There are many connections between the properties of the potentials $H$ and those of analytic functions $f$ of a complex variable (cf. also [[Analytic function|Analytic function]]; [[Harmonic function|Harmonic function]]). One such connection, related to the previous example, concerns singularities and uses the generating function for zonal harmonics to construct reciprocal integral transforms connecting $H$ with $f$. The following fact is immediate (see [[#References|[a3]]], [[#References|[a8]]]). Let $\{ a _ { n } \} _ { n = 0 } ^ { \infty }$ be a sequence of real constants for which $\operatorname {lim} \operatorname {sup}_{n \rightarrow \infty} | a _ { n } | ^ { 1 / n } = 1$. Consider the associated harmonic and analytic functions $H ( r , \theta ) = \sum _ { n = 0 } ^ { \infty } a _ { n } H _ { n } ( r , \theta )$ and $f ( z ) = \sum _ { n = 0 } ^ { \infty } a _ { n } z ^ { n }$, which are regular for $r = | z | < 1$. Then the boundary point $( 1 , \theta _ { 0 } )$ is a singularity of $H ( r , \theta )$ if and only if the boundary point $z = \operatorname { exp } ( i \theta _ { 0 } )$ is a singularity of $f ( z )$. Thus, the singularities of solutions of a singular partial differential equation are characterized in terms of those of associated analytic functions and vice versa.
  
From the 1950s onwards, an extensive literature has developed using integral transform methods to study solutions of large classes of multi-variable partial differential equations. The analysis is based on the theory of analytic and harmonic functions in several variables. Zonal harmonics play an important role in axially symmetric problems in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130130/z13013044.png" /> (see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a5]]]).
+
From the 1950s onwards, an extensive literature has developed using integral transform methods to study solutions of large classes of multi-variable partial differential equations. The analysis is based on the theory of analytic and harmonic functions in several variables. Zonal harmonics play an important role in axially symmetric problems in $\mathbf{R} ^ { 3 }$ (see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a5]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Begher,   R.P. Gilbert,   "Transmutations, transformations and kernel functions" , ''Monographs and Surveys in Pure and Applied Math.'' , '''58–59''' , Pitman (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Bergman,   "Integral operators in the theory of linear partial differential equations" , Springer (1963)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R.P. Gilbert,   "Function theoretic methods in partial differential equations" , ''Math. in Sci. and Engin.'' , '''54''' , Acad. Press (1969)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> O.D. Kellogg,   "Foundations of potential theory" , F. Ungar (1929)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Kracht,   E. Kreyszig,   "Methods of complex analysis in partial differential equations with applications" , Wiley/Interscience (1988)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> W.D. MacMillan,   "The theory of the potential" , Dover (1958)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> P.M. Morse,   H. Feshbach,   "Methods of theoretical physics" , '''1–2''' , McGraw-Hill (1953)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> G. Szegö,   "On the singularities of real zonal harmonic series" ''J. Rat. Mech. Anal.'' , '''3''' (1954) pp. 561–564</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> H. Begher, R.P. Gilbert, "Transmutations, transformations and kernel functions" , ''Monographs and Surveys in Pure and Applied Math.'' , '''58–59''' , Pitman (1992)</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> S. Bergman, "Integral operators in the theory of linear partial differential equations" , Springer (1963) {{MR|0239239}} {{MR|1532808}} {{MR|0180735}} {{MR|0141880}} {{ZBL|0209.40002}} {{ZBL|0176.08501}} {{ZBL|0121.07802}} {{ZBL|0093.28701}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> R.P. Gilbert, "Function theoretic methods in partial differential equations" , ''Math. in Sci. and Engin.'' , '''54''' , Acad. Press (1969) {{MR|0241789}} {{ZBL|0187.35303}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> O.D. Kellogg, "Foundations of potential theory" , F. Ungar (1929) {{MR|0222317}} {{MR|1522134}} {{ZBL|0152.31301}} {{ZBL|0053.07301}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> M. Kracht, E. Kreyszig, "Methods of complex analysis in partial differential equations with applications" , Wiley/Interscience (1988) {{MR|0941372}} {{ZBL|0644.35005}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> W.D. MacMillan, "The theory of the potential" , Dover (1958)</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> P.M. Morse, H. Feshbach, "Methods of theoretical physics" , '''1–2''' , McGraw-Hill (1953) {{MR|0059774}} {{ZBL|0051.40603}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> G. Szegö, "On the singularities of real zonal harmonic series" ''J. Rat. Mech. Anal.'' , '''3''' (1954) pp. 561–564</td></tr></table>

Latest revision as of 16:11, 11 February 2024

zonal harmonic polynomials

Zonal harmic polynomials are spherical harmonic polynomials (cf. also Spherical harmonics) that assume constant values on circles centred on an axis of symmetry. They characterize single-valued harmonic functions on simply-connected domains with rotational symmetry.

To be specific, one introduces the spherical coordinates $( r , \theta , \varphi )$ as $x _ { 1 } = r \operatorname { sin } \theta \operatorname { cos } \varphi$, $x _ { 2 } = r \operatorname { sin } \theta \operatorname{sin} \phi$, $x _ { 3 } = r \operatorname { cos } \theta$, where $( x _ { 1 } , x _ { 2 } , x _ { 3 } ) \in \mathbf{R} ^ { 3 }$. The zonal harmonics $H _ { n }$ are the polynomial solutions of the Laplace equation

\begin{equation*} \left[ \partial _ { r r } + \frac { 2 } { r } \partial _ { r } + \frac { 1 } { r ^ { 2 } } \partial _ { \theta \theta } + \frac { \operatorname { ctan } \theta } { r ^ { 2 } } \partial _ { \theta } + \frac { 1 } { r ^ { 2 } \operatorname { sin } ^ { 2 } \theta } \partial _ { \varphi \varphi } \right] H = 0 \end{equation*}

that are axially symmetric (i.e. independent of the angle $\varphi$). They can be expressed in terms of Legendre polynomials $P_n$ of degree $n$, as $H _ { n } ( r , \theta ) = r ^ { n } P _ { n } ( \operatorname { cos } \theta )$ for $n = 0,1 , \dots$, and form a complete orthogonal set of functions in $L ^ { 2 } [ D ]$, where $D$: $r \leq r_0$. The $H _ { n }$ vanish on cones that divide a sphere centred at the origin into $n$ zones, hence the name zonal harmonics. The $H _ { n }$ are sometimes referred to as solid zonal harmonics and the $P_n$ as surface zonal harmonics.

Applications.

Two types of applications arise in classical potential theory (see [a4], [a6], [a7]).

In the first, one determines the potential in a sphere from its boundary values $H ( r _ { 0 } , \theta )$. By specifying appropriate regularity conditions, the orthogonality of the Legendre polynomials is used to expand $H ( r _ { 0 } , \theta )$ as the Fourier–Legendre series $\sum _ { n = 0 } ^ { \infty } a _ { n } n_{0} ^ { n } P _ { n } ( \operatorname { cos } \theta )$. The potential in the sphere is recovered as $H ( r , \theta )$. The exterior boundary value problem is formulated by means of the Kelvin transformation $H ( r , \theta ) \rightarrow ( 1 / r ) H ( 1 / r ^ { 2 } , \theta )$. The potential between two concentric spheres is determined by combining solutions of the interior and the exterior problems.

In the second, one determines the potential at points in space from its values on a segment of the symmetry axis. The solution relies on the fact that along this axis the zonal harmonics $H _ { n } ( r , 0 ) = r ^ { n }$, $n = 0,1 , \dots$. Thus, if $H ( r , 0 ) = \sum _ { n = 0 } ^ { \infty } a _ { n } H _ { n } ( r , 0 )$, then $H ( r , \theta ) = \sum _ { n = 0 } ^ { \infty } a _ { n } H _ { n } ( r , \theta )$ for $r < r_{0}$, where $r_0$ is the radius of convergence of the Taylor series.

Relation with analytic functions.

There are many connections between the properties of the potentials $H$ and those of analytic functions $f$ of a complex variable (cf. also Analytic function; Harmonic function). One such connection, related to the previous example, concerns singularities and uses the generating function for zonal harmonics to construct reciprocal integral transforms connecting $H$ with $f$. The following fact is immediate (see [a3], [a8]). Let $\{ a _ { n } \} _ { n = 0 } ^ { \infty }$ be a sequence of real constants for which $\operatorname {lim} \operatorname {sup}_{n \rightarrow \infty} | a _ { n } | ^ { 1 / n } = 1$. Consider the associated harmonic and analytic functions $H ( r , \theta ) = \sum _ { n = 0 } ^ { \infty } a _ { n } H _ { n } ( r , \theta )$ and $f ( z ) = \sum _ { n = 0 } ^ { \infty } a _ { n } z ^ { n }$, which are regular for $r = | z | < 1$. Then the boundary point $( 1 , \theta _ { 0 } )$ is a singularity of $H ( r , \theta )$ if and only if the boundary point $z = \operatorname { exp } ( i \theta _ { 0 } )$ is a singularity of $f ( z )$. Thus, the singularities of solutions of a singular partial differential equation are characterized in terms of those of associated analytic functions and vice versa.

From the 1950s onwards, an extensive literature has developed using integral transform methods to study solutions of large classes of multi-variable partial differential equations. The analysis is based on the theory of analytic and harmonic functions in several variables. Zonal harmonics play an important role in axially symmetric problems in $\mathbf{R} ^ { 3 }$ (see [a1], [a2], [a3], [a5]).

References

[a1] H. Begher, R.P. Gilbert, "Transmutations, transformations and kernel functions" , Monographs and Surveys in Pure and Applied Math. , 58–59 , Pitman (1992)
[a2] S. Bergman, "Integral operators in the theory of linear partial differential equations" , Springer (1963) MR0239239 MR1532808 MR0180735 MR0141880 Zbl 0209.40002 Zbl 0176.08501 Zbl 0121.07802 Zbl 0093.28701
[a3] R.P. Gilbert, "Function theoretic methods in partial differential equations" , Math. in Sci. and Engin. , 54 , Acad. Press (1969) MR0241789 Zbl 0187.35303
[a4] O.D. Kellogg, "Foundations of potential theory" , F. Ungar (1929) MR0222317 MR1522134 Zbl 0152.31301 Zbl 0053.07301
[a5] M. Kracht, E. Kreyszig, "Methods of complex analysis in partial differential equations with applications" , Wiley/Interscience (1988) MR0941372 Zbl 0644.35005
[a6] W.D. MacMillan, "The theory of the potential" , Dover (1958)
[a7] P.M. Morse, H. Feshbach, "Methods of theoretical physics" , 1–2 , McGraw-Hill (1953) MR0059774 Zbl 0051.40603
[a8] G. Szegö, "On the singularities of real zonal harmonic series" J. Rat. Mech. Anal. , 3 (1954) pp. 561–564
How to Cite This Entry:
Zonal harmonics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zonal_harmonics&oldid=13900
This article was adapted from an original article by Peter A. McCoy (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article