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− | A polynomial with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h0465201.png" /> as variables that satisfies the [[Laplace equation|Laplace equation]]. Any harmonic polynomial may be represented as the sum of homogeneous harmonic polynomials. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h0465202.png" />, there are only two linearly independent homogeneous harmonic polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h0465203.png" /> — for example, the real and the imaginary part of the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h0465204.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h0465205.png" />, the number of linearly independent homogeneous polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h0465206.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h0465207.png" />. In the general case — <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h0465208.png" /> — the number of linearly independent homogeneous harmonic polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h0465209.png" /> is | + | {{TEX|done}} |
| + | A polynomial with $x_1,\ldots,x_n$ as variables that satisfies the [[Laplace equation|Laplace equation]]. Any harmonic polynomial may be represented as the sum of homogeneous harmonic polynomials. If $n=2$, there are only two linearly independent homogeneous harmonic polynomials of degree $m$ — for example, the real and the imaginary part of the expression $(x_1+ix_2)^m$. If $n=3$, the number of linearly independent homogeneous polynomials of degree $m$ is $2m+1$. In the general case — $n\geq2$ — the number of linearly independent homogeneous harmonic polynomials of degree $m$ is |
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− | <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/h/h046/h046520/h04652010.png" /></td> </tr></table>
| + | $$K_n^m-K_n^{m-2},\quad m\geq2,$$ |
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| where | | where |
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− | <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/h/h046/h046520/h04652011.png" /></td> </tr></table>
| + | $$K_n^m=\frac{n(n+1)\ldots(n+m-1)}{m!}$$ |
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− | is the number of permutations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652012.png" /> objects taken <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652013.png" /> at a time with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652014.png" /> repetitions. The homogeneous harmonic polynomials, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652015.png" />, are also known as [[Spherical functions|spherical functions]] (in particular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652016.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652017.png" />, one may write, in spherical coordinates | + | is the number of permutations of $n$ objects taken $m$ at a time with $m$ repetitions. The homogeneous harmonic polynomials, $V_m(x)$, are also known as [[Spherical functions|spherical functions]] (in particular if $n=3$). If $n=3$, one may write, in spherical coordinates |
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− | <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/h/h046/h046520/h04652018.png" /></td> </tr></table>
| + | $$V_m(x)=r^mY_m(\theta,\phi),$$ |
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− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652020.png" /> is a spherical function of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652021.png" />. | + | where $r=\sqrt{x_1^2+x_2^2+x_3^2}$ and $Y_m(\theta,\phi)$ is a spherical function of degree $m$. |
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| ====References==== | | ====References==== |
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| A finite linear combination of [[Harmonics|harmonics]]. Real-valued harmonic polynomials can be represented in the form | | A finite linear combination of [[Harmonics|harmonics]]. Real-valued harmonic polynomials can be represented in the form |
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− | <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/h/h046/h046520/h04652022.png" /></td> </tr></table>
| + | $$\sum_{k=1}^NA_k\sin(\omega_kx+\phi_k)$$ |
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− | for a given natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652023.png" />, non-negative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652024.png" />, and real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652027.png" />. Complex-valued harmonic polynomials can be represented in the form | + | for a given natural number $N$, non-negative $A_k$, and real $\omega_k$, $\phi_k$, $k=1,\ldots,N$. Complex-valued harmonic polynomials can be represented in the form |
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− | <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/h/h046/h046520/h04652028.png" /></td> </tr></table>
| + | $$\sum_{k=-m}^nc_ke^{i\omega_kx}$$ |
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− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652030.png" /> are natural numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652031.png" /> is real and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046520/h04652033.png" />, are complex. Harmonic polynomials are the simplest almost-periodic functions (cf. [[Almost-periodic function|Almost-periodic function]]). | + | where $n$ and $m$ are natural numbers, $\omega_k$ is real and the $c_k$, $k=-m,-m+1,\ldots,n$, are complex. Harmonic polynomials are the simplest almost-periodic functions (cf. [[Almost-periodic function|Almost-periodic function]]). |
A polynomial with $x_1,\ldots,x_n$ as variables that satisfies the Laplace equation. Any harmonic polynomial may be represented as the sum of homogeneous harmonic polynomials. If $n=2$, there are only two linearly independent homogeneous harmonic polynomials of degree $m$ — for example, the real and the imaginary part of the expression $(x_1+ix_2)^m$. If $n=3$, the number of linearly independent homogeneous polynomials of degree $m$ is $2m+1$. In the general case — $n\geq2$ — the number of linearly independent homogeneous harmonic polynomials of degree $m$ is
$$K_n^m-K_n^{m-2},\quad m\geq2,$$
where
$$K_n^m=\frac{n(n+1)\ldots(n+m-1)}{m!}$$
is the number of permutations of $n$ objects taken $m$ at a time with $m$ repetitions. The homogeneous harmonic polynomials, $V_m(x)$, are also known as spherical functions (in particular if $n=3$). If $n=3$, one may write, in spherical coordinates
$$V_m(x)=r^mY_m(\theta,\phi),$$
where $r=\sqrt{x_1^2+x_2^2+x_3^2}$ and $Y_m(\theta,\phi)$ is a spherical function of degree $m$.
References
[1] | S.L. Sobolev, "Partial differential equations of mathematical physics" , Pergamon (1964) (Translated from Russian) MR0178220 Zbl 0123.06508 |
[2] | A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) MR104888 |
[3] | M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) MR0106366 Zbl 0084.30903 |
E.D. Solomentsev
A finite linear combination of harmonics. Real-valued harmonic polynomials can be represented in the form
$$\sum_{k=1}^NA_k\sin(\omega_kx+\phi_k)$$
for a given natural number $N$, non-negative $A_k$, and real $\omega_k$, $\phi_k$, $k=1,\ldots,N$. Complex-valued harmonic polynomials can be represented in the form
$$\sum_{k=-m}^nc_ke^{i\omega_kx}$$
where $n$ and $m$ are natural numbers, $\omega_k$ is real and the $c_k$, $k=-m,-m+1,\ldots,n$, are complex. Harmonic polynomials are the simplest almost-periodic functions (cf. Almost-periodic function).