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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN"
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<title>SHTOOLS - Complex spherical harmonics</title>
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<p class="dir"><a class="dir2" href="conventions.html">Real spherical harmonics</a></p>
<h1>Complex Spherical Harmonics</h1>
<h2>"Geodesy" 4π-normalized complex spherical harmonics</h2>
<p>Any complex square-integrable function can be expressed in spherical harmonics as</p>
<img class="eqinline" src="images/f_c.png" alt="f_c.png" width=222 height=48>
<p>where f<sub>l</sub><sup>m</sup> is the complex spherical harmonic coefficient, Y<sub>l</sub><sup>m</sup> is the complex spherical harmonic function, position on the sphere is represented in terms of co-latitude θ and longitude φ,and <i>l</i> and <i>m</i> are the spherical harmonic degree and order, respectively. The complex spherical harmonics are defined as</p>
<img class="eqinline" src="images/ylm_complex.png" alt="ylm_complex.png" width=185 height=19>
<p>where the normalized associated Legendre functions, as used in geodesy and gravity applications (calculated by the routine <tt>PlmBar</tt> with <tt>cnorm=1</tt>), are given by</p>
<img class="eqinline" src="images/plmbar_complex.png" alt="plmbar_complex.png" width=251 height=49>
<p>with the following definition for the unnormalized associated Legendre functions:</p>
<img class="eqinline" src="images/plmdef.png" alt="plmdef.png" width=254 height=82>
<p>The normalized associated Legendre functions are orthogonal for a given value of <i>m</i>,</p>
<img class="eqinline" src="images/plmnorm_c.png" alt="plmnorm_c.png" width=196 height=41>.
<p>The normalized Legendre functions (including the orthonormalized and Schmidt semi-normalized functions) are efficiently calculating using the algorithm of <i>Holmes and Featherstone</i> (2002, J. Geodesy, 76, 279-299) that is accurate to approximately degree 2800. Note that the above definition of the associated Legendre functions does not include the Condon-Shortley phase of <tt>(-1)<sup>m</sup></tt> that is often employed in the physics and seismology communities (e.g., <i>Varsalovich et al., </i>1988; <i>Dahlen and Tromp,</i> 1998). This can be included by specifying the optional argument <tt>CSPHASE=-1</tt> in most of the Legendre and spherical harmonic routines.</p>
<p> The complex spherical harmonics possess the following symmetry relationship for positive and negative angular orders,</p>
<img class="eqinline" src="images/ylm_complex_sym.png" alt="ylm_complex_sym.png" width=209 height=18>
<p>and satisfy the orthogonality relationship</p>
<img class="eqinline" src="images/ylm_complex_norm.png" alt="ylm_complex_norm.png" width=276 height=37>.
<p>Using this relationship, it is straightforward to show that the spherical harmonic coefficients of a function can be calculated by the integral</p>
<img class="eqinline" src="images/flm_c.png" alt="flm_c.png" width=228 height=37>.
<p>The cross-power of two functions <i>f</i> and <i>g</i> is equal to</p>
<img class="eqinline" src="images/totalpower_c.png" alt="totalpower_c.png" width=256 height=45>,
<p>where the "cross-power spectrum" of the two functions is given by</p>
<img class="eqinline" src="images/crosspower_c.png" alt="crosspower_c.png" width=151 height=49>.
<p>If the functions <i>f</i> and <i>g</i> have a zero mean, then S<sub>fg</sub> is the contribution to the covariance as a function of spherical harmonic degree.</p>
<p>If a function defined on the sphere is entirely real, then the <a href="conventions.html">real</a> and complex spherical harmonic coefficients are related by</p>
<img class="eqinline" src="images/real_complex.png" alt="real_complex.png" width=267 height=59>
<h2>Other complex normalization conventions</h2>
<p class="nomarginbot">The above 4π normalization scheme is the default for all routines in SHTOOLS with the exception of <tt>MakeMagGrid2D</tt>, which uses the standard geomagnetism Schmidt semi-normalization. By specifying an optional parameter <tt>norm</tt>, the default normalization can be changed in most of the routines (e.g., <tt>SHGLQ</tt>, <tt>MakeGridGLQ</tt>, <tt>SHExpandGLQ</tt>, <tt>SHExpandDH</tt>, <tt>MakeGrid2D</tt>, <tt>MakeGridPoint</tt>, <tt>SHMultiply</tt> and <tt>SHExpandLSQ</tt>). The following normalizations are permitted:</p>
<ul>
<li> <tt>norm=1</tt>: Geodesy 4π normalization (default, unless stated otherwise).</li>
<li> <tt>norm=2</tt>: Schmidt semi-normalization.</li>
<li> <tt>norm=3</tt>: Unnormalized.</li>
<li> <tt>norm=4</tt>: Orthonormalized.</li>
</ul>
<p>Explicit expressions for the Legendre functions and real spherical harmonic normalizations are the following:</p>
<p class="underline">Geodesy 4π</p>
<img class="eqinline" src="images/plmbar_complex.png" alt="plmbar_complex.png" width=251 height=49><br>
<img class="eqinline" src="images/plmnorm_c.png" alt="plmnorm_c.png" width=196 height=41><br>
<img class="eqinline" src="images/ylm_complex_norm.png" alt="ylm_complex_norm.png" width=276 height=37><br>
<img class="eqinline" src="images/crosspower_c.png" alt="crosspower_c.png" width=151 height=49>
<p class="underline">Schmidt semi-normalized</p>
<img class="eqinline" src="images/plmschmidt_c.png" alt="plmschmidt_c.png" width=196 height=49><br>
<img class="eqinline" src="images/plmschmidtnorm_c.png" alt="plmschmidtnorm_c.png" width=231 height=41><br>
<img class="eqinline" src="images/ylmschmidt_c.png" alt="ylmschmidt_c.png" width=306 height=37><br>
<img class="eqinline" src="images/crosspower_schmidt_c.png" alt="crosspower_schmidt_c.png" width=211 height=49>
<p class="underline">Unnormalized</p>
<img class="eqinline" src="images/plmunnormalized_c.png" alt="plmunnormalized_c.png" width=118 height=18><br>
<img class="eqinline" src="images/plmunnormnorm_c.png" alt="plmunnormnorm_c.png" width=270 height=41><br>
<img class="eqinline" src="images/ylmunnorm_c.png" alt="ylmunnorm_c.png" width=365 height=38><br>
<img class="eqinline" src="images/crosspower_unnorm_c.png" alt="crosspower_unnorm_c.png" width=266 height=49>
<p class="underline">Orthonormalized</p>
<img class="eqinline" src="images/plmon_c.png" alt="plmon_c.png" width=252 height=49><br>
<img class="eqinline" src="images/plmonnorm_c.png" alt="plmonnorm_c.png" width=209 height=41><br>
<img class="eqinline" src="images/ylmorthonorm_c.png" alt="ylmorthonorm_c.png" width=258 height=37><br>
<img class="eqinline" src="images/crosspower_ortho_c.png" alt="crosspower_ortho_c.png" width=176 height=49>
<h2>Further information</h2>
<p class="nomarginbot">Further information regarding spherical harmonic functions can be found at the following web sites:</p>
<ul>
<li><a href="http://mathworld.wolfram.com/SphericalHarmonic.html">Mathworld - Spherical Harmonic</a></li>
<li><a href="http://en.wikipedia.org/wiki/Spherical_harmonics">Wikipedia - Spherical harmonics</a></li>
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