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We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave ${\mathrm{e}}^{i\widehat{s}·\overrightarrow{v}}$ in terms of spherical harmonics ${\left\{Y_{\ell ,m}\left(\widehat{s}\right)\right\}}_{\left|m\right|\le \ell \le \infty }$. We consider the truncated series where the summation is performed over the $(\ell ,m)$’s satisfying $\left|m\right|\le \ell \le L$. We prove that if $v=|\overrightarrow{v}|$ is large enough, the truncated series gives rise to an error lower than $ϵ$ as soon as $L$ satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K{ϵ}^{-\delta }{v}^{\gamma }\right)v^{\frac{1}{3}}$ where $W$ is the Lambert function and $C,K,\delta ,\gamma$ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results...

We perform a complete study of the truncation error of the Gegenbauer series. This series yields an expansion of the Green kernel of the Helmholtz equation, $\frac{{\mathrm{e}}^{i|\overrightarrow{u}-\overrightarrow{v}|}}{4\pi i|\overrightarrow{u}-\overrightarrow{v}|}$, which is the core of the Fast Multipole Method for the integral equations. We consider the truncated series where the summation is performed over the indices $\ell \le L$. We prove that if $v=|\overrightarrow{v}|$ is large enough, the truncated series gives rise to an error lower than $ϵ$ as soon as $L$ satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K\left(\alpha \right){ϵ}^{-\delta }{v}^{\gamma }\right)v^{\frac{1}{3}}$ where $W$ is the Lambert function, $K\left(\alpha \right)$ depends only on $\alpha =\frac{|\overrightarrow{u}|}{|\overrightarrow{v}|}$ and $C,\delta ,\gamma$ are...

We perform a complete study of the truncation error of the Gegenbauer series. This series yields an expansion of the Green kernel of the Helmholtz equation, $\frac{{\mathrm{e}}^{i|\overrightarrow{u}-\overrightarrow{v}|}}{4\pi i|\overrightarrow{u}-\overrightarrow{v}|}$, which is the core of the Fast Multipole Method for the integral equations. We consider the truncated series where the summation is performed over the indices $\ell \le L$. We prove that if $v=|\overrightarrow{v}|$ is large enough, the truncated series gives rise to an error lower than as soon as satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K\left(\alpha \right){ϵ}^{-\delta }{v}^{\gamma }\right)v^{\frac{1}{3}}$ where  is the Lambert function, $K\left(\alpha \right)$ depends only on $\alpha =\frac{|\overrightarrow{u}|}{|\overrightarrow{v}|}$ and...

We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave ${\mathrm{e}}^{i\widehat{s}·\overrightarrow{v}}$ in terms of spherical harmonics ${\left\{Y_{\ell ,m}\left(\widehat{s}\right)\right\}}_{\left|m\right|\le \ell \le \infty }$. We consider the truncated series where the summation is performed over the $(\ell ,m)$'s satisfying $\left|m\right|\le \ell \le L$. We prove that if $v=|\overrightarrow{v}|$ is large enough, the truncated series gives rise to an error lower than as soon as satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K{ϵ}^{-\delta }{v}^{\gamma }\right)v^{\frac{1}{3}}$ where is the Lambert function and $C,K,\delta ,\gamma$ are pure positive constants. Numerical experiments show that this asymptotic is optimal....