Appendix
1.1 Prosthaphaeresis formulae
$$\begin{aligned} \cos (m x)\cos (n x)= & {} \frac{1}{2}\left[ \cos ((m-n)x)+ \cos ((m+n)x)\right] \end{aligned}$$
(182)
$$\begin{aligned} \sin (m x)\sin (n x)= & {} \frac{1}{2}\left[ \cos ((m-n)x)- \cos ((m+n)x)\right] \end{aligned}$$
(183)
$$\begin{aligned} \sin (m x)\cos (n x)= & {} \frac{1}{2}\left[ \sin ((m-n)x)+\sin ((m+n)x)\right] \end{aligned}$$
(184)
$$\begin{aligned} \cos (m x)\sin (n x)= & {} \frac{1}{2}\left[ \sin ((m+n)x)-\sin ((m-n)x)\right] \end{aligned}$$
(185)
1.2 Series relations
$$\begin{aligned}{} & {} \sum \limits _{k=0}^{l_{max}} A_k\cos ((k-1)\nu )=\sum \limits _{k=-1}^{l_{max}-1} A_{k+1}\cos (k\nu )\nonumber \\{} & {} \quad =A_0 \cos (\nu )+A_1 +\sum \limits _{k=1}^{l_{max}-1}A_{k+1}\cos (k\nu ) \end{aligned}$$
(186)
$$\begin{aligned}{} & {} \sum \limits _{k=0}^{l_{max}} A_k\cos ((k+1)\nu )=\sum \limits _{k=1}^{l_{max}+1} A_{k-1}\cos (k\nu ) \end{aligned}$$
(187)
$$\begin{aligned}{} & {} \sum \limits _{k=0}^{l_{max}} B_k\sin ((k-1)\nu )=\sum \limits _{k=-1}^{l_{max}-1} B_{k+1}\sin (k\nu )=-B_0\sin (\nu )+\sum \limits _{k=1}^{l_{max}-1} B_{k+1} \sin (k\nu ) \nonumber \\\end{aligned}$$
(188)
$$\begin{aligned}{} & {} \sum \limits _{k=0}^{l_{max}} B_k\sin ((k+1)\nu )=\sum \limits _{k=1}^{l_{max}+1} B_{k-1}\sin (k\nu ) \end{aligned}$$
(189)
1.2.1 Application to cosine series
$$\begin{aligned}{} & {} \sum \limits _{m=0}^\infty A_m \cos (m\nu )\sum \limits _{n=0}^\infty B_n \cos (n\nu )\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_m B_n \cos (m\nu ) \cos (n\nu )\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_m B_n \frac{1}{2}\left[ \cos ((m-n)\nu )+ \cos ((m+n)\nu )\right] \nonumber \\{} & {} \quad =\frac{1}{2}\left[ A_{0}B_{0}+\sum \limits _{p=0}^{\infty } A_{p} B_p \right] \nonumber \\{} & {} \quad +\frac{1}{2}\sum \limits _{p=1}^{\infty } \left[ \sum \limits _{n=0}^\infty B_n A_{p+n} +\sum \limits _{n=p}^{\infty } B_n A_{-p+n} + \sum \limits _{n=0}^{p} B_n A_{p-n} \right] \cos (p\nu ) \end{aligned}$$
(190)
1.2.2 Application to sine times cosine series
$$\begin{aligned}{} & {} \sum \limits _{m=0}^\infty B_m \sin (m x) \sum \limits _{n=0}^\infty A_n \cos (n x)\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty B_{m} A_n \sin (m x) \cos (n x)\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty B_{m} A_n \frac{1}{2}\left[ \sin ((m-n) x)+\sin ((m+n) x)\right] \nonumber \\{} & {} \quad =\frac{1}{2}\sum \limits _{p=1}^\infty \left[ \sum _{n=0}^\infty A_n B_{p+n} -\sum \limits _{n=p}^\infty A_n B_{-p+n} +\sum \limits _{n=0}^p A_n B_{p-n} \right] \sin (p x) \end{aligned}$$
(191)
1.2.3 Application to cosine times sine series
$$\begin{aligned}{} & {} \sum \limits _{m=0}^\infty A_m \cos (m\nu )\sum \limits _{n=0}^\infty B_n \sin (n\nu )\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_m B_n \cos (m\nu ) \sin (n\nu )\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_m B_n \frac{1}{2}\left[ \sin ((m+n)\nu )-\sin ((m-n)\nu )\right] \nonumber \\{} & {} \quad =\frac{1}{2}\sum \limits _{p=1}^{\infty }\left[ -\sum \limits _{n=0}^\infty B_n A_{p+n} +\sum \limits _{n=p}^{\infty } B_n A_{-p+n} + \sum \limits _{n=0}^{p} B_n A_{p-n} \right] \sin (p\nu ) \end{aligned}$$
(192)
1.2.4 Application to sine series
$$\begin{aligned}{} & {} \sum \limits _{m=0}^\infty A_{m} \sin (m\nu )\sum \limits _{n=0}^\infty B_n \sin (n\nu )\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_{m} B_n \sin (m\nu ) \sin (n\nu )\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_{m} B_n \frac{1}{2}\left[ \cos ((m-n)\nu )- \cos ((m+n)\nu )\right] \nonumber \\{} & {} \quad =\frac{1}{2}\left[ - A_{0} B_0+\sum \limits _{p=0}^{\infty } A_{p} B_p \right] \nonumber \\{} & {} \quad +\frac{1}{2}\sum \limits _{p=1}^{\infty } \left[ \sum \limits _{n=0}^\infty B_n A_{p+n} +\sum \limits _{n=p}^{\infty } B_n A_{-p+n} - \sum \limits _{n=0}^{p} B_n A_{p-n} \right] \cos (p\nu ) \end{aligned}$$
(193)
1.3 Indefinite integrals
\(n \ne 0\)
$$\begin{aligned} \int \sin (n x)dx= & {} -\frac{1}{n} \cos (n x) +C \end{aligned}$$
(194)
$$\begin{aligned} \int \cos (n x)dx= & {} \frac{1}{n} \sin (n x) +C \end{aligned}$$
(195)
$$\begin{aligned} \int x \sin (n x) dx= & {} \frac{1}{n^2} \sin (n x) - \frac{x}{n}\cos (n x) + C \end{aligned}$$
(196)
$$\begin{aligned} \int x \cos (n x) dx= & {} \frac{1}{n^2} \cos (n x) + \frac{x}{n}\sin (n x) + C \end{aligned}$$
(197)
$$\begin{aligned} \int x^2 \cos (n x) dx= & {} \frac{x^2}{n}\sin (n x) -\frac{2}{n^3}\sin (n x) +\frac{2x}{n^2}\cos (nx) + C \end{aligned}$$
(198)
1.3.1 Cosine times cosine series
$$\begin{aligned}{} & {} \cos (\nu )\sum \limits _{n=0}^\infty A_n \cos (n\nu )=\sum \limits _{n=0}^\infty \frac{1}{2} A_n \left[ \cos ((1-n)\nu )+ \cos ((1+n)\nu )\right] \nonumber \\{} & {} \quad =\sum \limits _{n=0}^\infty \frac{1}{2} A_n \cos ((n-1)\nu )+\sum \limits _{n=0}^\infty \frac{1}{2} A_n \cos ((1+n)\nu ) \end{aligned}$$
(199)
$$\begin{aligned}{} & {} \int \cos (\nu )\sum \limits _{n=0}^\infty A_n \cos (n\nu ) d \nu \nonumber \\{} & {} \quad =C+\sum \limits _{n=0}^\infty \frac{1}{2} {\left\{ \begin{array}{ll} \frac{A_n}{n-1} \sin ((n-1)\nu ),\ \textrm{if} \ n\ne 1\\ A_1 \nu ,\ \textrm{if} \ n=1\\ \end{array}\right. } +\sum \limits _{n=0}^\infty \frac{1}{2} \frac{A_n}{n+1} \sin ((n+1)\nu )\nonumber \\{} & {} \quad = C + \frac{A_1}{2} \nu + \left[ A_0 +\frac{A_2}{2} \right] \sin (\nu )+\frac{1}{2}\sum \limits _{n=2}^\infty \frac{A_{n+1}+A_{n-1}}{n} \sin (n\nu ) \end{aligned}$$
(200)
1.3.2 Cosine times sine series
$$\begin{aligned}{} & {} \cos (\nu )\sum \limits _{n=0}^\infty B_n \sin (n\nu )=\sum \limits _{n=0}^\infty \frac{1}{2} B_n \left[ \sin ((1+n)\nu )-\sin ((1-n)\nu )\right] \nonumber \\{} & {} \quad =\sum \limits _{n=0}^\infty \frac{1}{2} B_n \sin ((1+n)\nu )+\sum \limits _{n=0}^\infty \frac{1}{2} B_n \sin ((n-1)\nu ) \end{aligned}$$
(201)
$$\begin{aligned}{} & {} \int \cos (\nu )\sum \limits _{n=0}^\infty B_n \sin (n\nu ) d\nu \nonumber \\{} & {} \quad =C-\sum \limits _{n=0}^\infty \frac{1}{2} \frac{B_n}{1+n} \cos ((1+n)\nu ) -\sum \limits _{n=0}^\infty \frac{1}{2} {\left\{ \begin{array}{ll} \frac{B_n}{n-1} \cos ((n-1)\nu ),\ \textrm{if} \ n\ne 1\\ 0 ,\ \textrm{if} \ n=1\\ \end{array}\right. }\nonumber \\{} & {} \quad = C -\frac{B_2}{2}\cos (\nu )-\frac{1}{2}\sum \limits _{n=2}^\infty \frac{B_{n-1}+B_{n+1}}{n} \cos (n\nu ) \end{aligned}$$
(202)
1.3.3 Sine times cosine series
$$\begin{aligned}{} & {} \sin (\nu )\sum \limits _{n=0}^\infty A_n \cos (n\nu )=\sum \limits _{n=0}^\infty \frac{1}{2} A_n \left[ \sin ((1-n)\nu )+\sin ((1+n)\nu )\right] \nonumber \\{} & {} \quad =-\sum \limits _{n=0}^\infty \frac{1}{2} A_n \sin ((n-1)\nu )+\sum \limits _{n=0}^\infty \frac{1}{2} A_n \sin ((1+n)\nu ) \end{aligned}$$
(203)
$$\begin{aligned}{} & {} \int \sin (\nu )\sum \limits _{n=0}^\infty A_n \cos (n\nu ) d \nu \nonumber \\{} & {} \quad =C+\sum \limits _{n=0}^\infty \frac{1}{2} {\left\{ \begin{array}{ll} \frac{A_n}{n-1} \cos ((n-1)\nu ),\ \textrm{if} \ n\ne 1\\ 0 ,\ \textrm{if} \ n=1\\ \end{array}\right. }-\sum \limits _{n=0}^\infty \frac{1}{2} \frac{A_n}{1+n} \cos ((1+n)\nu )\nonumber \\{} & {} \quad = C + \left[ -A_0 +\frac{A_2}{2} \right] \cos (\nu )+\frac{1}{2}\sum \limits _{n=2}^\infty \frac{A_{n+1}-A_{n-1}}{n}\cos (n\nu ) \end{aligned}$$
(204)
1.3.4 Sine times sine series
$$\begin{aligned}{} & {} \sin (\nu )\sum \limits _{n=0}^\infty B_n \sin (n\nu )=\sum \limits _{n=0}^\infty \frac{1}{2} B_n \left[ \cos ((1-n)\nu )- \cos ((1+n)\nu )\right] \nonumber \\{} & {} \quad =\sum \limits _{n=0}^\infty \frac{1}{2} B_n \cos ((n-1)\nu )-\sum \limits _{n=0}^\infty \frac{1}{2} B_n \cos ((1+n)\nu ) \end{aligned}$$
(205)
$$\begin{aligned}{} & {} \int \sin (\nu )\sum \limits _{n=0}^\infty B_n \sin (n\nu ) d \nu \nonumber \\{} & {} \quad =C+\sum \limits _{n=0}^\infty \frac{1}{2} {\left\{ \begin{array}{ll}\frac{B_n}{n-1} \sin ((n-1)\nu ),\ \textrm{if} \ n\ne 1\\ B_1 \nu ,\ \textrm{if} \ n=1\\ \end{array}\right. }-\sum \limits _{n=0}^\infty \frac{1}{2} \frac{B_n}{1+n} \sin ((1+n)\nu )\nonumber \\{} & {} \quad = C + \frac{B_1}{2} \nu + \frac{B_2}{2}\sin (\nu )+\frac{1}{2}\sum \limits _{n=2}^\infty \frac{B_{n+1}-B_{n-1}}{n}\sin (n\nu ) \end{aligned}$$
(206)
1.3.5 Product of cosine series
$$\begin{aligned}{} & {} \sum \limits _{m=0}^\infty A_m \cos (m\nu )\sum \limits _{n=0}^\infty B_n \cos (n\nu )\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_m B_n \cos (m\nu ) \cos (n\nu ) \nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_m B_n \frac{1}{2}\left[ \cos ((m-n)\nu )+ \cos ((m+n)\nu )\right] \end{aligned}$$
(207)
$$\begin{aligned}{} & {} \int \sum \limits _{m=0}^\infty A_m \cos (m\nu ) \sum \limits _{n=0}^\infty B_n \cos (n\nu ) d \nu = C+\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_{m} B_n\nonumber \\{} & {} \quad \times \frac{1}{2} \left[ {\left\{ \begin{array}{ll} \frac{\sin ((m-n)\nu )}{m-n},\ \textrm{if} \ m-n \ne 0\\ \nu ,\ \textrm{if} \ m-n=0\\ \end{array}\right. }+ {\left\{ \begin{array}{ll}\frac{\sin ((m+n)\nu )}{m+n},\ \textrm{if} \ m+n \ne 0\\ \nu ,\ \textrm{if} \ m+n=0\\ \end{array}\right. }\right] \nonumber \\{} & {} \quad =C+\frac{1}{2}\left[ A_{0}B_{0}+\sum \limits _{p=0}^{\infty } A_{p} B_p \right] \nu \nonumber \\{} & {} \quad +\frac{1}{2}\sum \limits _{p=1}^{\infty }\frac{1}{p} \left[ \sum \limits _{n=0}^\infty B_n A_{p+n} +\sum \limits _{n=p}^{\infty } B_n A_{-p+n} + \sum \limits _{n=0}^{p} B_n A_{p-n} \right] \sin (p\nu ) \end{aligned}$$
(208)
1.3.6 Product of cosine and sine series
$$\begin{aligned}{} & {} \sum \limits _{m=0}^\infty A_m \cos (m\nu )\sum \limits _{n=0}^\infty B_n \sin (n\nu )\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_m B_n \cos (m\nu ) \sin (n\nu )\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_m B_n \frac{1}{2}\left[ \sin ((m+n)\nu )-\sin ((m-n)\nu )\right] \end{aligned}$$
(209)
$$\begin{aligned}{} & {} \int \sum \limits _{m=0}^\infty A_m \cos (m\nu ) \sum \limits _{n=0}^\infty B_n \sin (n\nu ) d\nu =C+ \sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_m B_n \frac{1}{2}\times \nonumber \\{} & {} \quad \left[ {\left\{ \begin{array}{ll} \frac{\cos ((m-n)\nu )}{m-n},\ \textrm{if} \ m-n \ne 0\\ 0 ,\ \textrm{if} \ m-n=0\\ \end{array}\right. }- {\left\{ \begin{array}{ll}\frac{\cos ((m+n)\nu )}{m+n},\ \textrm{if} \ m+n \ne 0\\ 0 ,\ \textrm{if} \ m+n=0\\ \end{array}\right. }\right] \nonumber \\{} & {} \quad =C+\frac{1}{2}\sum \limits _{p=1}^{\infty }\frac{1}{p} \left[ \sum \limits _{n=0}^\infty B_n A_{p+n} -\sum \limits _{n=p}^{\infty } B_n A_{-p+n} - \sum \limits _{n=0}^{p} B_n A_{p-n} \right] \cos (p\nu ) \end{aligned}$$
(210)
1.3.7 Product of sine and cosine series
$$\begin{aligned}{} & {} \sum \limits _{m=0}^\infty B_{m} \sin (m\nu )\sum \limits _{n=0}^\infty A_n \cos (n\nu )\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty B_{m} A_n \sin (m\nu ) \cos (n\nu )\nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty B_{m} A_n \frac{1}{2}\left[ \sin ((m-n)\nu )+\sin ((m+n)\nu )\right] \end{aligned}$$
(211)
$$\begin{aligned}{} & {} \int \sum \limits _{m=0}^\infty B_{m} \sin (m\nu )\ \sum \limits _{n=0}^\infty A_n \cos (n\nu ) d\nu =C+\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty -B_{m} A_n \frac{1}{2} \times \nonumber \\{} & {} \quad \left[ {\left\{ \begin{array}{ll} \frac{\cos ((m-n)\nu )}{m-n},\ \textrm{if} \ m-n \ne 0\\ 0 ,\ \textrm{if} \ m-n=0\\ \end{array}\right. }+ {\left\{ \begin{array}{ll}\frac{\cos ((m+n)\nu )}{m+n},\ \textrm{if} \ m+n \ne 0\\ 0 ,\ \textrm{if} \ m+n=0\\ \end{array}\right. }\right] \nonumber \\{} & {} \quad =C+\frac{1}{2}\sum \limits _{p=1}^{\infty }\frac{1}{p} \left[ -\sum \limits _{n=0}^\infty A_n B_{p+n} +\sum \limits _{n=p}^{\infty } A_n B_{-p+n} - \sum \limits _{n=0}^{p} A_n B_{p-n} \right] \cos (p\nu )\qquad \quad \end{aligned}$$
(212)
1.3.8 Product of sine series
$$\begin{aligned}{} & {} \sum \limits _{m=0}^\infty A_{m} \sin (m\nu )\sum \limits _{n=0}^\infty B_n \sin (n\nu ) \nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_{m} B_n \sin (m\nu ) \sin (n\nu ) \nonumber \\{} & {} \quad =\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_{m} B_n \frac{1}{2}\left[ \cos ((m-n)\nu )- \cos ((m+n)\nu )\right] \end{aligned}$$
(213)
$$\begin{aligned}{} & {} \int \sum \limits _{m=0}^\infty A_{m} \sin (m\nu ) \sum \limits _{n=0}^\infty B_n \sin (n\nu ) d \nu =C+\sum \limits _{m=0}^\infty \sum \limits _{n=0}^\infty A_{m} B_n \frac{1}{2}\times \nonumber \\{} & {} \quad \left[ {\left\{ \begin{array}{ll} \frac{\sin ((m-n)\nu )}{m-n},\ \textrm{if} \ m-n \ne 0\\ \nu ,\ \textrm{if} \ m-n=0\\ \end{array}\right. }- {\left\{ \begin{array}{ll}\frac{\sin ((m+n)\nu )}{m+n},\ \textrm{if} \ m+n \ne 0\\ \nu ,\ \textrm{if} \ m+n=0\\ \end{array}\right. }\right] \nonumber \\{} & {} \quad =C+\frac{1}{2}\left[ - A_{0} B_0+\sum \limits _{p=0}^{\infty } A_{p} B_p \right] \nu + \nonumber \\{} & {} \quad \frac{1}{2}\sum \limits _{p=1}^{\infty }\frac{1}{p} \left[ \sum \limits _{n=0}^\infty B_n A_{p+n} +\sum \limits _{n=p}^{\infty } B_n A_{-p+n} - \sum \limits _{n=0}^{p} B_n A_{p-n} \right] \sin (p\nu ) \end{aligned}$$
(214)
1.3.9 Indefinite integrals applied to useful fourier series
First the integral of a Fourier series:
$$\begin{aligned} \int \sum \limits _{n_x=0}^\infty f_{n_x} d \nu = A^\prime _0 \nu +\sum \limits _{n_x=1}^\infty \frac{A^\prime _{n_x} \sin (n_x \nu )}{n_x} -\frac{B^\prime _{n_x} \cos (n_x \nu )}{n_x} \end{aligned}$$
(215)
The integral of cosine of true anomaly times a Fourier series:
$$\begin{aligned}{} & {} \int \cos (\nu )\sum \limits _{n_x=0}^\infty f_{n_x} d\nu =\frac{A^\prime _1}{2} \nu -\frac{B^\prime _2}{2}\cos (\nu )+ \left[ A^\prime _0 +\frac{A^\prime _2}{2} \right] \sin (\nu ) \nonumber \\{} & {} \quad +\frac{1}{2}\sum \limits _{n=2}^\infty -\frac{B^\prime _{n-1}+B^\prime _{n+1}}{n} \cos (n\nu ) +\frac{1}{2}\sum \limits _{n=2}^\infty \frac{A^\prime _{n+1}+A^\prime _{n-1}}{n} \sin (n\nu ) \;. \end{aligned}$$
(216)
The integral of sine of true anomaly times a Fourier series:
$$\begin{aligned} \int \sin (\nu )\sum \limits _{n_x=0}^\infty f_{n_x} d\nu =\frac{B^\prime _1}{2} \nu + \left[ -A^\prime _0 +\frac{A^\prime _2}{2} \right] \cos (\nu )+ \frac{B^\prime _2}{2}\sin (\nu )\nonumber \\ +\frac{1}{2}\sum \limits _{n=2}^\infty \frac{A^\prime _{n+1}-A^\prime _{n-1}}{n}\cos (n\nu )+\frac{1}{2}\sum \limits _{n=2}^\infty \frac{B^\prime _{n+1}-B^\prime _{n-1}}{n}\sin (n\nu ) \;. \end{aligned}$$
(217)
The integral of \(1/(1+e\cos (\nu ))\) times a Fourier series:
$$\begin{aligned}{} & {} \int \sum \limits _{m=0}^\infty \beta _{m} \sum \limits _{n_x=0}^\infty f_{n_x} d\nu =\frac{1}{2}\left[ b^1_{0}A^\prime _{0}+\sum \limits _{p=0}^{\infty } b^1_{p} A^\prime _p \right] \nu \nonumber \\{} & {} \quad +\frac{1}{2}\sum \limits _{p=1}^{\infty }\frac{1}{p} \left[ \sum \limits _{n=0}^\infty A^\prime _n b^1_{p+n} +\sum \limits _{n=p}^{\infty } A^\prime _n b^1_{-p+n} + \sum \limits _{n=0}^{p} A^\prime _n b^1_{p-n} \right] \sin (p\nu ) \nonumber \\{} & {} \quad +\frac{1}{2}\sum \limits _{p=1}^{\infty }\frac{1}{p} \left[ \sum \limits _{n=0}^\infty B^\prime _n b^1_{p+n} -\sum \limits _{n=p}^{\infty } B^\prime _n b^1_{-p+n} - \sum \limits _{n=0}^{p} B^\prime _n b^1_{p-n} \right] \cos (p\nu ) \;. \end{aligned}$$
(218)
The integral of \(\cos (\nu )/(1+e\cos (\nu ))\) times a Fourier series:
$$\begin{aligned}{} & {} \int \sum \limits _{m=0}^\infty \beta _{c,m} \sum \limits _{n_x=0}^\infty f_{n_x} d\nu =\frac{1}{2}\left[ B^1_{c,0}A^\prime _{0}+\sum \limits _{p=0}^{\infty } B^1_{c,p} A^\prime _p \right] \nu \nonumber \\{} & {} \quad +\frac{1}{2}\sum \limits _{p=1}^{\infty }\frac{1}{p} \left[ \sum \limits _{n=0}^\infty B^\prime _n B^1_{c,p+n} -\sum \limits _{n=p}^{\infty } B^\prime _n B^1_{c,-p+n} - \sum \limits _{n=0}^{p} B^\prime _n B^1_{c,p-n} \right] \cos (p\nu ) \nonumber \\{} & {} \quad +\frac{1}{2}\sum \limits _{p=1}^{\infty }\frac{1}{p} \left[ \sum \limits _{n=0}^\infty A^\prime _n B^1_{c,p+n} +\sum \limits _{n=p}^{\infty } A^\prime _n B^1_{c,-p+n} + \sum \limits _{n=0}^{p} A^\prime _n B^1_{c,p-n} \right] \sin (p\nu ) \;. \end{aligned}$$
(219)
The integral of \(\sin (\nu )/(1+e\cos (\nu ))\) times a Fourier series:
$$\begin{aligned}{} & {} \int \sum \limits _{m=0}^\infty \beta _{s,m} \sum \limits _{n_x=0}^\infty f_{n_x} d\nu =\frac{1}{2}\left[ - B^1_{s,0} B^\prime _0+\sum \limits _{p=0}^{\infty } B^1_{s,p} B^\prime _p \right] \nu \nonumber \\{} & {} \quad +\frac{1}{2}\sum \limits _{p=1}^{\infty }\frac{1}{p} \left[ -\sum \limits _{n=0}^\infty A^\prime _n B^1_{s,p+n} +\sum \limits _{n=p}^{\infty } A^\prime _n B^1_{s,-p+n} - \sum \limits _{n=0}^{p} A^\prime _n B^1_{s,p-n} \right] \cos (p\nu ) \nonumber \\{} & {} \quad +\frac{1}{2}\sum \limits _{p=1}^{\infty }\frac{1}{p} \left[ \sum \limits _{n=0}^\infty B^\prime _n B^1_{s,p+n} +\sum \limits _{n=p}^{\infty } B^\prime _n B^1_{s,-p+n} - \sum \limits _{n=0}^{p} B^\prime _n B^1_{s,p-n} \right] \sin (p\nu ) \;. \end{aligned}$$
(220)
1.4 Definite integrals
\(n \ne 0\)
$$\begin{aligned}{} & {} \int \limits _0^{2\pi }\sin (n x)dx= 0 \end{aligned}$$
(221)
$$\begin{aligned}{} & {} \int \limits _0^{2\pi }\cos (n x)dx= 0 \end{aligned}$$
(222)
$$\begin{aligned}{} & {} \int \limits _0^{2\pi } x \sin (n x) dx= - \frac{2\pi }{n} \end{aligned}$$
(223)
$$\begin{aligned}{} & {} \int \limits _0^{2\pi } x \cos (n x) dx= 0 \end{aligned}$$
(224)
$$\begin{aligned}{} & {} \int \limits _0^{2\pi } x^2 \cos (n x) dx= \frac{4\pi }{n^2} \end{aligned}$$
(225)
$$\begin{aligned}{} & {} \int \limits _0^{2\pi } \cos (n x) \cos (m x) dx={\left\{ \begin{array}{ll} 0, \; n \ne m \\ \pi , \; n = m \end{array}\right. } \end{aligned}$$
(226)
$$\begin{aligned}{} & {} \int \limits _0^{2\pi } \sin (n x) \sin (m x) dx={\left\{ \begin{array}{ll} 0, \; n \ne m \\ \pi , \; n = m \end{array}\right. } \end{aligned}$$
(227)
$$\begin{aligned}{} & {} \int \limits _0^{2\pi } \sin (n x) \cos (m x) dx=0 \end{aligned}$$
(228)
1.5 Fourier series expansions of powers of the conic equation
Recursion relations for Fourier series expansions of powers of the conic equation from Scheeres (1992) and Scheeres (2012):
$$\begin{aligned}{} & {} \left( \frac{1}{1+e\cos \nu } \right) ^n=\sum \limits _{m=0}^\infty b_m^n\cos (m\nu ) \end{aligned}$$
(229)
$$\begin{aligned}{} & {} b_0^n=\frac{\sqrt{1-e^2}}{\left( 1-e^2 \right) ^n} f_0^n \end{aligned}$$
(230)
$$\begin{aligned}{} & {} b_k^n=(-1)^k 2 \left( \frac{e}{2}\right) ^k \frac{\sqrt{1-e^2}}{\left( 1-e^2 \right) ^n} f_k^n \end{aligned}$$
(231)
$$\begin{aligned}{} & {} f_k^{n+1}={\left\{ \begin{array}{ll} \frac{(n-k)!(n+k)!}{(n!)^2}\sum \limits _{l=0}^{[(n-k)/2]} \frac{n!}{l!(l+k)!(n-k-2l)!} \left( \frac{e}{2} \right) ^{2l} &{} n+1>k\\ \frac{n-k}{n} \left( 1-e^2\right) f_k^n+2f_{k-1}^{n+1} &{} n+1 \le k \end{array}\right. } \end{aligned}$$
(232)
$$\begin{aligned}{} & {} f^1_k=\left( \frac{2}{1+\sqrt{1-e^2}} \right) ^k \end{aligned}$$
(233)
where the notation [x] means to round down to the nearest integer.
