### 0.1 Spherical Harmonics


**Definition 0.1.1 **  The **Legendre polynomials** $P_\ell(x)$ are a set of polynomials defined on the interval $-1\leq x \leq 1$ for $\ell \in \{0,1,2,\dots\}$ by the following expression.
\[P_\ell(x) = \frac{1}{2^\ell}\]

**Definition 0.1.2 **  The **associated Legendre polynomials** $P_\ell^m(x)$ are a set of polynomials defined on the interval $-1\leq x \leq 1$ for $\ell \in \{0,1,2,3,\dots\}$ and $m \in \{-\ell,-\ell+1,\dots,0,\dots,\ell-1,\ell\}$ by the following expression.
\[P_\ell^m(x) = \frac{(-1)^{m}}{2^\ell\ell!}(1-x^2)^{m/2}\frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell\]

**Definition 0.1.3 **  The **spherical harmonics** $Y_{\ell,m}(\theta,\phi)$ are a set of spherical functions defined for $\ell \in \{0,1,2,3,\dots\}$ and $m \in \{-\ell,-\ell+1,\dots,0,\dots,\ell-1,\ell\}$ that forms an orthonormal bases for the set of complex spherical functions.
\[Y_{\ell,m}(\theta,\phi) = \sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} P_\ell^m(\cos\theta)e^{im\phi}\]

**Result 0.1.4 **  **Properties of Spherical Harmonics:** The spherical harmonics $Y_{\ell,m}$ and $Y_{\ell, -m}$ are related by the following relation,
\[Y_{\ell,-m}(\theta,\phi) = (-1)^{m}Y_{\ell,m}^*(\theta, \phi)\]
Spherical harmonics from an orthonormal basis with the following orthogonality and normalization conditions,
\[\int_0^{2\pi}\int_0^\pi Y^*_{\ell,m'}(\theta,\phi)Y_{\ell,m}(\theta,\phi)  \sin\theta\ d\theta\ d\phi = \delta_{\ell',\ell}\delta_{m',m}\]
Spherical harmonics span the set of all complex spherical functions as show by the following completeness relation,
\[\sum_{\ell = 0}^\infty \sum_{m=-\ell}^\ell Y_{\ell,m}^*(\theta',\phi')Y_{\ell,m}(\theta,\phi) = \delta(\phi-\phi')\delta(\cos\theta-\cos\theta')\]

**Table 0.1.5 **  **Table of Spherical Harmoincs**
\[Y_{0,0} = \frac{1}{\sqrt{4\pi}}\]
\[Y_{1,1}  = -\sqrt{\frac{3}{8\pi}}\sin\theta\ e^{i\phi}\]
\[Y_{1,0} = \sqrt{\frac{3}{4\pi}}\cos\theta\]
\[Y_{2,2} = \frac{1}{4}\sqrt{\frac{15}{2\pi}} \sin^2\theta\ e^{2i\phi}\]
\[Y_{2,1} = -\sqrt{\frac{15}{8\pi}}\sin\theta\cos\theta\ e^{i\phi}\]
\[Y_{2,0} = \sqrt{\frac{5}{4\pi}}\left(\frac{3}{2}\cos^2\theta - \frac{1}{2}\right)\]