\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 318, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/318\hfil Upper bounds for the number of limit cycles]
{Upper bounds for the number of limit cycles of polynomial differential systems}

\author[S. Ellaggoune, S. Badi\hfil EJDE-2016/318\hfilneg]
{Selma Ellaggoune, Sabrina Badi}

\address{Selma Ellaggoune \newline
Laboratory of Applied Mathematics and Modeling,
University 8 Mai 1945,
Guelma, Algeria}
\email{sellaggoune@gmail.com}

\address{Sabrina Badi \newline
Laboratory of Applied Mathematics and Modeling,
 University 8 Mai 1945,
Guelma, Algeria}
\email{badisabrina@yahoo.fr}

\thanks{Submitted October 1, 2016. Published December 14, 2016.}
\subjclass[2010]{34C25, 34C29, 37G15}
\keywords{Limit cycle; Li\'enard differential equation; averaging method}

\begin{abstract}
 For $\varepsilon$ small we consider the number of
 limit cycles of the polynomial differential system
 $$
 \dot{x}=y-f_1(x,y)y,  \quad \dot{y}=-x-g_2(x,y)-f_2(x,y)y,
 $$
 where $f_1(x,y)=\varepsilon f_{11}(x,y)+\varepsilon^2f_{12}(x,y)$,
 $g_2(x,y)=\varepsilon g_{21}(x,y)+\varepsilon^2 g_{22}(x,y)$ and
 $f_2(x,y)=\varepsilon f_{21}(x,y)+\varepsilon^2 f_{22}(x,y)$
 where $f_{1i}, f_{2i}, g_{2i}$ have degree $l, n,m$ respectively for
 each $i=1,2$.
 We provide an accurate upper bound of the maximum number of limit cycles that
 this class of systems can have bifurcating from the periodic
 orbits of the linear center $\dot{x}=y, \dot{y}=-x$ using the
 averaging theory of first and second order. We give an example for which
 this bound is reached.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks

\section{Introduction}

One of the main problems in the theory of ordinary differential
equations is the study of their limit cycles, their existence,
their number and their stability. A limit cycle of a differential
equation is a periodic orbit in the set of all isolated periodic
orbits of the differential equation. The second part of the 16th
Hilbert's problem (see \cite{H}) is related to the least upper bound
on the number of limit cycles of polynomial vector fields having a
fixed degree. These years many papers have studied the limit
cycles of planar polynomial differential systems. In this paper we
will try to give a partial answer to this problem for the class of
polynomial differential systems given by
\begin{equation}\label{eq1}
\dot{x}=y-f_1(x,y)y, \quad \dot{y}=-x-g_2(x,y)-f_2(x,y)y,
\end{equation}
where $f_1(x,y)=\varepsilon f_{11}(x,y)+\varepsilon^2f_{12}(x,y)$,
$g_2(x,y)=\varepsilon g_{21}(x,y)+\varepsilon^2 g_{22}(x,y)$ and
$f_2(x,y)=\varepsilon f_{21}(x,y)+\varepsilon^2 f_{22}(x,y)$
where$f_{1i}, f_{2i}$ and $g_{2i}$ have degree $l, n$ and $m$
respectively for each $i=1,2$ and $\varepsilon$ is a small
parameter. Note that when $f_1(x,y)=0, g_2(x,y)=0 $ and
$f_2(x,y)=f(x)$ these systems coincide with the
generalized polynomial Li\'enard differential systems
\begin{equation}\label{eq2}
\dot{x}=y, \quad   \dot{y}=-x-f(x)y,
\end{equation}
where $f(x)$ is a polynomial in the variable $x$ of
degree $n$.

In 1977 Lins, de Melo and Pugh \cite{LDMP} studied the classical polynomial
Li\'enard differential system \eqref{eq2} and stated
the following conjecture:
\begin{quote}
If $f(x)$ has degree $n\geq 1$, then \eqref{eq2} has at most
$[\frac{n}{2}]$ limit cycles.
\end{quote}
Then they proved this conjecture for $n=1, 2$.
The conjecture for $n=3$ has been proved recently by Chengzi and Llibre
\cite{LL}. For more information see \cite{Lv}.

Many of the results on the limit cycles of polynomial differential
systems have been obtained by considering limit cycles which
bifurcate from a single degenerate singular point, that are so
called {\it small amplitude limit cycles}, see for instance
\cite{LnL}. We denote by $\hat{H}(m,n)$ the maximum number of
small amplitude limit cycles for systems of the form \eqref{eq2}.
The values of $\hat{H}(m,n)$ give a lower bound for the maximum
number $H(m,n)$ (i.e. The Hilbert number) of limit cycles that the
differential equation \eqref{eq2} with $m$ and $n$ fixed can have.
For more information about the Hilbert's 16th problem and related
topics see \cite{I}.

In \cite{LV} the authors ise the averaging theory of
first and second order to study the system
\begin{equation}\label{eq3}
\begin{gathered}
\dot{x} = y -\varepsilon(g_{11}(x)+f_{11}(x)y)-\varepsilon^2 (g_{12}(x)+f_{12}(x)y),
 \\
\dot{y} = -x -\varepsilon(g_{21}(x)+f_{21}(x)y)-\varepsilon^2 (g_{22}(x)+f_{22}(x)y),
\end{gathered}
\end{equation}
where $g_{1i}, f_{1i}, g_{2i}, f_{2i}$ have degree $l, k, m, n$ respectively
for each $i=1,2$, and $\varepsilon$ is a small
parameter. They provided an accurate upper bound of the maximum number of
limit cycles that the above system can have bifurcating from the periodic
orbits of the linear center $\dot{x} = y, \dot{y} = -x$.

In this article, first we consider the more general system
\begin{equation}\label{eq4}
\begin{gathered}
\dot{x} = y -\varepsilon(f_{11}(x,y)y), \\
\dot{y} = -x -\varepsilon(g_{21}(x,y)+f_{21}(x,y)y), \\
\end{gathered}
\end{equation}
where $f_{11}, g_{21}$ and $f_{21}$ have degree $l, m$ and $n$
respectively, and $\varepsilon$ is a small parameter. We obtain
the following result.

\begin{theorem} \label{thm1.1}
 For $|\varepsilon|$ sufficiently small, the maximum number of
 limit cycles of the generalized polynomial differential
 system \eqref{eq4} bifurcating from the periodic orbits of the
 linear center $\dot{x}=y, \dot{y}=-x$ using the averaging theory
 of first order is
 $[\frac{n}{2}]$.
\end{theorem}

The proof of the above theorem  is given in section 3.

Secondly we consider the system
\begin{equation}\label{eq5}
\begin{gathered}
\dot{x} = y -\varepsilon(f_{11}(x,y)y)-\varepsilon^2(f_{12}(x,y)y), \\
\dot{y} = -x -\varepsilon(g_{21}(x,y)+f_{21}(x,y)y)
-\varepsilon^2(g_{22}(x,y)+f_{22}(x,y)y),
\end{gathered}
\end{equation}
where $f_{11}$ and$f_{12}$ have degree $l$; $g_{21}$ and $g_{22}$
have degree $m$; and $f_{21},f_{22}$ have degree $n$. Furthermore,
$\varepsilon$ is a small parameter.
We obtain the theorem bellow.

\begin{theorem} \label{thm1.2}
 For $|\varepsilon|$ sufficiently small, the maximum number of
 limit cycles of the generalized polynomial differential
 system \eqref{eq5} bifurcating from the periodic orbits of the
 linear center $\dot{x}=y, \dot{y}=-x$ using the averaging theory
 of second order is
 $$
\lambda=\max \{\lambda_1, \lambda_2+1, \lambda_3+2 \},
$$
 where
\begin{gather*}
\begin{aligned}
\lambda_1
=\max \Big\{&[\frac{O(m)+E(l)-1}{2}], [\frac{E(m)+O(l)-1}{2}],
 [\frac{O(n)+O(l)-2}{2}], \\
&  m-1,[\frac{E(n)}{2}], [\frac{E(m)+O(n)-1}{2}]\Big\},
\end{aligned}\\
\begin{aligned}
\lambda_2
=\max\Big\{&O(n)-1, [\frac{E(m)+O(n)-3}{2}],[\frac{O(m)+E(n)-3}{2}], \\
&[\frac{O(n)+O(l)-2}{2}],
l-1, E(m)-2, [\frac{E(m)+O(n)-3}{2}],\\
& [\frac{O(m)+E(l)-3}{2}], [\frac{E(n)+O(m)-3}{2}] \Big\},
\end{aligned}\\
\lambda_3=[\frac{E(n)+E(l)-4}{2}],
\end{gather*}
where $O(i)$ is the largest odd integer less than or
equal to $i$, $E(i)$ is the largest even integer less than or
equal to $i$ and $[\cdot]$ denotes the integer part function.
\end{theorem}


The proof of the above theorem is given in section 4.
The results that we shall use from the averaging theory of first
and second order for computing limit cycles are presented in
section 2.

\section{Averaging theory of first and second order}

The averaging theory of first and second orders was introduced to
study periodic orbits, which is summarized as follows.
Consider a differential system
\begin{equation}\label{e2.1)}
\dot{x}(t)=\varepsilon F_1(t, x)+\varepsilon^2F_2(t,
x)+\varepsilon^3R(t,x,\varepsilon),
\end{equation}
where $F_1, F_2: \mathbb{R}\times D\to
\mathbb{R}^n$, $R: \mathbb{R}\times D\times (-\varepsilon_{f}
,\varepsilon_{f})\to \mathbb{R}^n$ are continuous
functions, which are T-periodic in the first variable, and $D$ is
an open subset of $\mathbb{R}^n$. Assume that:
\begin{itemize}

\item[(i)] $F_1(t,\cdot)\in \mathcal{C}^1(D)$ for all $t\in \mathbb{R}$,
$F_1, F_2, R, D_{x}F_1$ are locally Lipschitz with respect
to $x$, and $R$ is differentiable with respect to $\varepsilon$.
We define
\begin{gather*}
F_{10}(z)=\frac{1}{T}\int_0^{T}F_1(s,z) ds,\\
F_{20}(z)=\frac{1}{T}\int_0^{T}[D_{z}F_1(s,z)y_1(s,z)+ F_2(s,z)]ds,
\end{gather*}
where
$$
y_1(s,z)=\int_0^{\theta}F_1(t,z)dt.
$$

\item[(ii)] For $V\subset D$ an open and bounded set and for each
$\varepsilon \in(-\varepsilon_{f} ,\varepsilon_{f})\setminus\{0\}$,
 there exists an $a_{\varepsilon}\in V$ such that
$F_{10}(a_{\varepsilon})+\varepsilon F_{20}(a_{\varepsilon})=0$
and $d_{B}(F_{10}+\varepsilon F_{20},V,a_{\varepsilon})\neq 0$.
Then, for $|\varepsilon|>0$ sufficiently small there exists a
T-periodic solution $\varphi(\cdot, \varepsilon)$ of the system
\eqref{e2.1)} such that
$\varphi(0,\varepsilon)=a_{\varepsilon}$.
\end{itemize}

The expression $d_{B}(F_{10}+\varepsilon
F_{20},V,a_{\varepsilon})\neq 0$ means that the Brouwer degree of
the function $F_{10}+\varepsilon F_{20}: V\to \mathbb{R}^n$ at the fixed point
$a_{\varepsilon}$ is not zero.
A sufficient condition for the inequality to be true is that the
Jacobian of the function $F_{10}+\varepsilon F_{20}$ at
$a_{\varepsilon}$ is not zero.

If $F_{10}$ is not identically zero, then the zeros of
$F_{10}+\varepsilon F_{20}$ are mainly those of $F_{10}$
for $\varepsilon$ sufficiently small. In this case the previous
result provides the averaging theory of first order.

If $F_{10}$ is identically zero and $F_{20}$ is not identically
zero, then the zeros of $F_{10}+\varepsilon F_{20}$ are mainly the
zeros of $F_{20}$ for $\varepsilon$ sufficiently small. In this
case the previous result provides the averaging theory of second
order.

For more information about the averaging theory see (\cite{SV},\cite{V}).

\section{Proof of Theorem \ref{thm1.1}}

We need the first order averaging theory, for
this we write system \eqref{eq4} in polar coordinates $(r,\theta)$ where
$x=r\cos(\theta)$, $y=r\sin(\theta)$, $r > 0$. In this way system
\eqref{eq4} is written in the standard form for applying the
averaging theory.
If we write
\begin{equation}\label{e3.1)}
\begin{gathered}
f_{11}(x,y)= \sum_{i+j=0}^la_{ij,1}x^{i}y^{j}, \quad
f_{21}(x,y)=\sum_{i+j=0}^na_{ij,2}x^{i}y^{j}, \\\
g_{21}(x,y)= \sum_{i+j=0}^m b_{ij,2}x^{i}y^{i},
\end{gathered}
\end{equation}
then system \eqref{eq4} becomes
\begin{equation} \label{e3.2)}
\begin{gathered}
\begin{aligned}
\dot{r}
&=-\varepsilon\Big(\sum_{i+j=0}^na_{ij,2}r^{i+j+1}
 \cos^{i}(\theta)\sin^{j+2}(\theta)
 + \sum_{i+j=0}^m b_{ij,2}r^{i+j}\cos^{i}(\theta)\sin^{j+1}(\theta)\\
&\quad +\sum_{i+j=0}^la_{ij,1}r^{i+j+1}\cos^{i+1}(\theta)\sin^{j+1}(\theta)\Big),
\end{aligned}\\
\begin{aligned}
\dot{\theta}
&=-1-\frac{1}{r}\Big[\varepsilon\Big(\sum_{i+j=0}^na_{ij,2}r^{i+j+1}
 \cos^{i+1}(\theta)\sin^{j+1}(\theta) \\
&\quad +\sum_{i+j=0}^m b_{ij,2}r^{i+j}\cos^{i+1}(\theta)\sin^{j}(\theta) \\
&\quad -\sum_{i+j=0}^la_{ij,1}r^{i+j+1}\cos^{i}(\theta)\sin^{j+2}(\theta)\Big)\Big].
\end{aligned}
\end{gathered}
\end{equation}
Now taking $\theta$ as the new independent variable, this system
becomes
$$
\frac{dr}{d\theta}=\varepsilon F_1(r,\theta)+O(\varepsilon^2),
$$
where
\begin{align*}
F_1(r,\theta)
&= \sum_{i+j=0}^na_{ij,2}r^{i+j+1}\cos^{i}(\theta)\sin^{j+2}(\theta)+
\sum_{i+j=0}^m b_{ij,2}r^{i+j}\cos^{i}(\theta)\sin^{j+1}(\theta)\\
&\quad +\sum_{i+j=0}^la_{ij,1}r^{i+j+1}\cos^{i+1}(\theta)\sin^{j+1}(\theta).
\end{align*}
Using the notation introduced in section $2$ we must calculate
$$
F_{10}(r)=\frac{1}{2\pi}\int_0^{2\pi}F_1(r,\theta)d\theta.
$$
Since
\[
\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+2}(\theta)d\theta
=\begin{cases}
 0 & \text{if $i$ odd or $j$ odd,}\\
\pi \alpha_{ij} & \text{if $i$ even, $j$ even,}
\end{cases}
\]
where $\alpha_{ij}$ is a constant, we obtain
\begin{equation}\label{e3.3)}
F_{10}(r)=\frac{1}{2}r\sum_{i+j=0}^na_{ij,2}\alpha_{ij}r^{i+j},
\end{equation}
where $i$ and $j$ are both even.

Then the polynomial $F_{10}(r)$ has at most $[\frac{n}{2}]$
positive roots, and we can choose the coefficients $\alpha_{ij}$
with $i$ even, $j$ even in such a way that $F_{10}(r)$ has
exactly $[\frac{n}{2}]$ simple positive roots, hence Theorem \ref{thm1.1}
is proved.

\section{Proof of Theorem \ref{thm1.2}}

For the proof we shall use the second
order averaging theory as it was stated in section 2. We write
$f_{11}, f_{21}$ and $g_{21}$ as in \eqref{e3.1)} and
\begin{gather*}
f_{12}(x,y)= \sum_{i+j=0}^lC_{ij,1}x^{i}y^{j}, \quad
f_{22}(x,y)= \sum_{i+j=0}^nc_{ij,2}x^{i}y^{j},  \\
g_{22}(x,y)=\sum_{i+j=0}^m d_{ij,2}x^{i}y^{i}.
\end{gather*}
Then system \eqref{eq5} in polar coordinates becomes
\begin{equation}\label{e4.1)}
\begin{gathered}
\dot{r}=-\varepsilon(A+\varepsilon B),\\
\dot{\theta}=-1 - \frac{\varepsilon}{r} (A_1+\varepsilon
B_1),
\end{gathered}
\end{equation}
where
\begin{gather*}
\begin{aligned}
A&= \sum_{i+j=0}^na_{ij,2}r^{i+j+1}\cos^{i}(\theta)sin^{j+2}(\theta)
+\sum_{i+j=0}^m b_{ij,2}r^{i+j}\cos^{i}(\theta)\sin^{j+1}(\theta)\\
&\quad +\sum_{i+j=0}^la_{ij,1}r^{i+j+1}\cos^{i+1}(\theta)\sin^{j+1}(\theta),
\end{aligned} \\
\begin{aligned}
B&= \sum_{i+j=0}^nc_{ij,2}r^{i+j+1}\cos^{i}(\theta)\sin^{j+2}(\theta)
+\sum_{i+j=0}^m d_{ij,2}r^{i+j}\cos^{i}(\theta)\sin^{j+1}(\theta)\\
&\quad +\sum_{i+j=0}^lC_{ij,1}r^{i+j+1}\cos^{i+1}(\theta)\sin^{j+1}(\theta),
\end{aligned} \\
\begin{aligned}
A_1&= \sum_{i+j=0}^na_{ij,2}r^{i+j+1}\cos^{i+1}(\theta)\sin^{j+1}(\theta)+
\sum_{i+j=0}^m b_{ij,2}r^{i+j}\cos^{i+1}(\theta)\sin^{j}(\theta)\\
&\quad -\sum_{i+j=0}^la_{ij,1}r^{i+j+1}\cos^{i}(\theta)\sin^{j+2}(\theta),
\end{aligned} \\
\begin{aligned}
B_1&= \sum_{i+j=0}^nc_{ij,2}r^{i+j+1}\cos^{i+1}(\theta)\sin^{j+1}(\theta)+
\sum_{i+j=0}^m d_{ij,2}r^{i+j}\cos^{i+1}(\theta)\sin^{j}(\theta)\\
&\quad -\sum_{i+j=0}^lC_{ij,1}r^{i+j+1}\cos^{i}(\theta)\sin^{j+2}(\theta).
\end{aligned}
\end{gather*}
Taking $\theta$ as the new independent variable, this system
becomes
$$
\frac{dr}{d\theta}=\varepsilon F_1(r,\theta)+\varepsilon ^2 F_2(r,\theta)
+O(\varepsilon^3),
$$
where
$$
F_1(r,\theta)=A,\quad
F_2(r,\theta)=B-\frac{1}{r}AA_1.
$$
To compute $F_{20}(r)$, we need that $F_{10}(r)$ be identically zero, which is
equivalent to $a_{ij,2}=0$ for $i$ even, $j$ even.

 Now we determine the corresponding function
$$
F_{20}(r)=\frac{1}{2\pi}\int_0^{2\pi}
\Big[\frac{d}{dr}F_1(r,\theta)y_1(r,\theta)+F_2(r,\theta)\Big]d\theta.
$$
First, we have
\begin{align*}
\frac{d}{dr}F_1(r,\theta)
&=\sum_{\substack{ i+j=0\\  \text{$i$ odd  or $j$  odd}}}^n
 (i+j+1)a_{ij,2}r^{i+j} \cos^{i}(\theta)\sin^{j+2}(\theta) \\
&\quad +\sum_{i+j=0}^m (i+j)b_{ij,2}r^{i+j-1}
 \cos^{i}(\theta)\sin^{j+1}(\theta) \\
&\quad +\sum_{i+j=0}^l(i+j+1)a_{ij,1}r^{i+j}
\cos^{i+1}(\theta)\sin^{j+1}(\theta),
\end{align*}
and we write
$$
y_1(r,\theta)=\int_0^{\theta}F_1(r,t)dt=y_1^1+y_1^2+y_1^3,
$$
so we obtain
\begin{align*}
y_1^1(r,t)
&=\int_0^{\theta} \sum_{i+j=0}^na_{ij,2}r^{i+j+1}cos^{i}(t)\sin^{j+2}(t)dt \\
&=a_{10,2}r^2\Big(\alpha_{110}\sin(\theta)+\alpha_{210}\sin(3\theta)\Big)+\dots \\
&\quad +a_{c_1e_1,2}r^{c_1+e_1+1}\Big(\alpha_{1c_1e_1}\sin(\theta)
 +\alpha_{2c_1e_1}\sin(3\theta)+\dots \\
&\quad +\alpha_{\frac{(c_1+e_1+2)+1}{2}c_1e_1}\sin((c_1+e_1+2)\theta)\Big) \\
&\quad +a_{01,2}r^2\Big(\alpha_{101}+\alpha_{201}\cos(\theta)
 +\alpha_{301}\cos(3\theta)\Big)+\dots \\
&\quad +a_{p_1q_1,2}r^{p_1+q_1+1}\Big(\alpha_{1p_1q_1}
 +\alpha_{2p_1q_1}\cos(\theta)+\alpha_{3p_1q_1}\cos(3\theta)+\dots \\
&\quad +\alpha_{\frac{(p_1+q_1+2)+3}{2}p_1q_1}\cos((p_1+q_1+2)\theta)\Big) \\
&\quad +a_{11,2}r^3\Big(\alpha_{111}+\alpha_{211}\cos(2\theta)
 +\alpha_{311}\cos(4\theta)\Big)+\dots \\
&\quad +a_{c_1q_1,2}r^{c_1+q_1+1}\Big(\alpha_{1c_1q_1}
+\alpha_{2c_1q_1}\cos(2\theta)+\alpha_{3c_1q_1}\cos(4\theta)+\dots  \\
&\quad +\alpha_{\frac{(c_1+q_1+2)+2}{2}c_1q_1}\cos((c_1+q_1+2)\theta)\Big),
\end{align*}
where $c_1$ is the greatest odd number and $e_1$ is the
greatest even number so that $c_1+e_1$ is less than or equal to $n$.
$p_1$ is the greatest even number and $q_1$ is the greatest
odd number so that $p_1+q_1$ is less than or equal to $n$.
$\alpha_{ijk}$ are real constants exhibited during the computation
of $\int_0^{\theta}\cos^{i}(t)\sin^{j+2}(t)dt$ for all $i$ and
$j$.

\begin{align*}
y_1^2(r,t)
&=\int_0^{\theta} \sum_{i+j=0}^m b_{ij,2}r^{i+j}cos^{i}(t)\sin^{j+1}(t)dt\\
&=b_{00,2}\Big(\tilde{\alpha}_{100}+\tilde{\alpha}_{200}\cos(\theta)\Big)
 +b_{02,2}r^2\Big(\tilde{\alpha}_{102}+\tilde{\alpha}_{202}\cos(\theta)
 +\tilde{\alpha}_{302}\cos(3\theta)\Big)\\
&\quad +\dots  +b_{p_2e_2,2}r^{p_2+e_2}\Big(\tilde{\alpha}_{1p_2e_2}
 +\tilde{\alpha}_{2p_2e_2}\cos(\theta)+\tilde{\alpha}_{3p_2e_2}\cos(3\theta)+\dots\\
&\quad  +\tilde{\alpha}_{(\frac{p_2+e_2}{2}+2)p_2e_2}cos((p_2+e_2+1)\theta)\Big)
 +b_{01,2}r\Big(\tilde{\alpha}_{101}\theta+\tilde{\alpha}_{201}\sin(2\theta)\Big)\\
&\quad +\dots  +b_{p_2q_2,2}r^{p_2+q_2}\Big(\tilde{\alpha}_{1p_2q_2}\theta
+\tilde{\alpha}_{2p_2q_2}\sin(2\theta)+\tilde{\alpha}_{3p_2q_2}\sin(4\theta)
 +\dots\\
&\quad +\tilde{\alpha}_{(\frac{p_2+q_2+3}{2})p_2q_2}\sin((p_2+q_2+1)\theta)\Big)
+b_{10,2}r\Big(\tilde{\alpha}_{110}+\tilde{\alpha}_{210}\cos(2\theta)\Big) \\
&\quad +b_{30,2}r^3\Big(\tilde{\alpha}_{130}+\tilde{\alpha}_{230}\cos(2\theta)
 +\tilde{\alpha}_{330}\cos(4\theta)\Big)+\dots
 +b_{c_2e_2,2} r^{c_2+e_2} \\
&\quad\times \Big(\tilde{\alpha}_{1c_2e_2}+\tilde{\alpha}_{2c_2e_2}\cos(2\theta)
 +\dots   +\tilde{\alpha}_{(\frac{c_2+e_2+1}{2}+1)c_2e_2}
 \cos((c_2+e_2+1)\theta)\Big) \\
&\quad +b_{11,2}r^2
 \Big(\tilde{\alpha}_{111}\sin(\theta)+\tilde{\alpha}_{211}\sin(3\theta)\Big)\\
&\quad +b_{13,2}r^4\Big(\tilde{\alpha}_{113}\sin(\theta)
+\tilde{\alpha}_{213}\sin(3\theta)+\tilde{\alpha}_{313}\sin(5\theta)\Big)+\dots \\
&\quad+ b_{c_2q_2,2}r^{c_2+q_2}\Big(\tilde{\alpha}_{1c_2q_2\sin(\theta)}
 +\tilde{\alpha}_{2c_2q_2}\sin(3\theta)+\dots \\
&\quad +\tilde{\alpha}_{(\frac{c_2+q_2+2}{2})c_2q_2}\sin((c_2+q_2+1)\theta)\Big),
\end{align*}
where $p_2$ is the greatest even number and $e_2$ is the
greatest even number so that $p_2+e_2$ is less than or equal
to $m$.
$c_2$ is the greatest odd number and $q_2$ is the greatest odd
number so that $c_2+q_2$ is less than or equal
to $m$.
$\tilde{\alpha}_{ijk}$ are real constants exhibited during the
computation of $\int_0^{\theta}\cos^{i}(t)\sin^{j+1}(t)dt$ for
all $i$ and $j$.

\begin{align*}
&y_1^3(r,t)\\
&=\int_0^{\theta}\sum_{i+j=0}^la_{ij,1}r^{i+j+1}cos^{i+1}(t)\sin^{j+1}(t)dt\\
&=a_{00,1}r\Big(\hat{\alpha}_{100}+\hat{\alpha}_{200}\cos(2\theta)\Big)+\dots
+a_{p_3e_3,1}r^{p_3+e_3+1}\Big(\hat{\alpha}_{1p_3e_3}
+\hat{\alpha}_{2p_3e_3}\cos(2\theta) \\
&\quad +\dots  +\hat{\alpha}_{(\frac{p_3+e_3+2}{2}+1)p_3e_3}\cos((p_3
 +e_3+2)\theta)\Big) \\
&\quad +a_{01,1}r^2\Big(\hat{\alpha}_{101}
 \sin(\theta)+\hat{\alpha}_{201}\sin(3\theta)\Big)+\dots \\
&\quad +a_{p_3q_3,1}r^{p_3+q_3+1}\Big(\hat{\alpha}_{1p_3q_3}
 \sin(\theta)+\hat{\alpha}_{2p_3q_3}\sin(3\theta)
 +\hat{\alpha}_{3p_3q_3}\sin(5\theta)+\dots \\
&\quad +\hat{\alpha}_{(\frac{p_3+q_3+3}{2})p_3q_3}
 \sin((p_3+q_3+2)\theta)\Big) \\
&\quad +a_{10,1}r^2\Big(\hat{\alpha}_{110}+\hat{\alpha}_{210}\cos(\theta)
 +\hat{\alpha}_{310}\cos(3\theta)\Big)+\dots \\
&\quad +a_{c_3e_3,1}r^{c_3+e_3+1}\Big(\hat{\alpha}_{1c_3e_3}
 +\hat{\alpha}_{2c_3e_3}\cos(\theta)+\hat{\alpha}_{3c_3e_3}\cos(3\theta)
+\dots \\
&\quad +\hat{\alpha}_{(\frac{c_3+e_3+1}{2}+2)c_3e_3}
 \cos((c_3+e_3+2)\theta)\Big)
 +a_{11,1}r^3\Big(\hat{\alpha}_{111}\theta+\hat{\alpha}_{211}\sin(4\theta)\Big)\\
&\quad +a_{13,1}r^5\Big(\hat{\alpha}_{113}\theta+\hat{\alpha}_{213}\sin(2\theta)
+\hat{\alpha}_{313}\sin(4\theta)+\hat{\alpha}_{413}\sin(6\theta)\Big)+\dots \\
&\quad +a_{c_3q_3,1}r^{c_3+q_3+1}\Big(\hat{\alpha}_{1c_3q_3}\theta
 +\hat{\alpha}_{2c_3q_3}\sin(2\theta)\\
&\quad +\hat{\alpha}_{3c_3q_3}\sin(4\theta)+\dots
 +\hat{\alpha}_{(c_3+q_3)c_3q_3}\sin((c_3+q_3+2)\theta)\Big),
\end{align*}
where $p_3$ is the greatest even number and $e_3$ is the
greatest even number so that $p_3+e_3$ is less than or equal to $l$,
$c_3$ is the greatest odd number and $q_3$ is the greatest odd
number so that $c_3+q_3$ is less than or equal to $l$,
$\hat{\alpha}_{ijk}$ are real constants exhibited during the
computation of $\int_0^{\theta}\cos^{i+1}(t)\sin^{j+1}(t)dt$ for
all $i$ and $j$.

Finally
\begin{align*}
&y_1(r,\theta)\\
&=a_{10,2}r^2\Big(\alpha_{110}\sin(\theta)+\alpha_{210}\sin(3\theta)\Big)+\dots
 +a_{c_1e_1,2}r^{c_1+e_1+1}\Big(\alpha_{1c_1e_1}\sin(\theta) \\
&\quad +\alpha_{2c_1e_1}\sin(3\theta)+\dots
+\alpha_{\frac{(c_1+e_1+2)+1}{2}c_1e_1}\sin((c_1+e_1+2)\theta)\Big)\\
&\quad +a_{01,2}r^2\Big(\alpha_{101}+\alpha_{201}\cos(\theta)
 +\alpha_{301}\cos(3\theta)\Big)+\dots \\
&\quad +a_{p_1q_1,2}r^{p_1+q_1+1}\Big(\alpha_{1p_1q_1}
 +\alpha_{2p_1q_1}\cos(\theta)+\alpha_{3p_1q_1}\cos(3\theta)+\dots \\
&\quad +\alpha_{\frac{(p_1+q_1+2)+3}{2}p_1q_1}\cos((p_1+q_1+2)\theta)\Big)\\
&\quad +a_{11,2}r^3\Big(\alpha_{111}+\alpha_{211}\cos(2\theta)
 +\alpha_{311}\cos(4\theta)\Big)+\dots\\
&\quad +a_{c_1q_1,2}r^{c_1+q_1+1}\Big(\alpha_{1c_1q_1}
 +\alpha_{2c_1q_1}\cos(2\theta)+\alpha_{3c_1q_1}\cos(4\theta)+\dots \\
&\quad +\alpha_{\frac{(c_1+q_1+2)+2}{2}c_1q_1} \cos((c_1+q_1+2)\theta)\Big)
\\
&\quad + b_{00,2}\Big(\tilde{\alpha}_{100}+\tilde{\alpha}_{200}
 \cos(\theta)\Big)+b_{02,2}r^2\Big(\tilde{\alpha}_{102}
 +\tilde{\alpha}_{202}\cos(\theta)\\
&\quad +\tilde{\alpha}_{302}\cos(3\theta)\Big)+\dots
+b_{p_2e_2,2}r^{p_2+e_2}\Big(\tilde{\alpha}_{1p_2e_2}
+\tilde{\alpha}_{2p_2e_2}\cos(\theta)+\tilde{\alpha}_{3p_2e_2}\cos(3\theta)+\dots\\
&\quad +\tilde{\alpha}_{(\frac{p_2+e_2}{2}+2)p_2e_2}cos((p_2+e_2+1)\theta)\Big)
+b_{01,2}r\Big(\tilde{\alpha}_{101}\theta+\tilde{\alpha}_{201}\sin(2\theta)\Big)
+\dots \\
&\quad +b_{p_2q_2,2}r^{p_2+q_2}\Big(\tilde{\alpha}_{1p_2q_2}\theta
 +\tilde{\alpha}_{2p_2q_2}\sin(2\theta)+\tilde{\alpha}_{3p_2q_2}
 \sin(4\theta)+\dots \\
&\quad +\tilde{\alpha}_{(\frac{p_2+q_2+3}{2})p_2q_2}
 \sin((p_2+q_2+1)\theta)\Big)+b_{10,2}r
\Big(\tilde{\alpha}_{110}+\tilde{\alpha}_{210}\cos(2\theta)\Big) \\
&\quad +b_{30,2}r^3\Big(\tilde{\alpha}_{130}+\tilde{\alpha}_{230}\cos(2\theta)
  +\tilde{\alpha}_{330}\cos(4\theta)\Big)+\dots \\
&\quad +b_{c_2e_2,2}r^{c_2+e_2}\Big(\tilde{\alpha}_{1c_2e_2}
 +\tilde{\alpha}_{2c_2e_2}\cos(2\theta)+\dots \\
&\quad  +\tilde{\alpha}_{(\frac{c_2+e_2+1}{2}+1)c_2e_2}
 \cos((c_2+e_2+1)\theta)\Big)+b_{11,2}r^2
 \Big(\tilde{\alpha}_{111}\sin(\theta)+\tilde{\alpha}_{211}\sin(3\theta)\Big) \\
&\quad +b_{13,2}r^4\Big(\tilde{\alpha}_{113}\sin(\theta)
 +\tilde{\alpha}_{213}\sin(3\theta)+\tilde{\alpha}_{313}\sin(5\theta)\Big)
 +\dots \\
&\quad +b_{c_2q_2,2}r^{c_2+q_2}\Big(\tilde{\alpha}_{1c_2q_2\sin(\theta)}
 +\tilde{\alpha}_{2c_2q_2}\sin(3\theta)+\dots \\
&\quad +\tilde{\alpha}_{(\frac{c_2+q_2+2}{2})c_2q_2}\sin((c_2+q_2+1)\theta)\Big)
 +a_{00,1}r\Big(\hat{\alpha}_{100}+\hat{\alpha}_{200}\cos(2\theta)\Big)+\dots \\
&\quad +a_{p_3e_3,1}r^{p_3+e_3+1}
 \Big(\hat{\alpha}_{1p_3e_3}+\hat{\alpha}_{2p_3e_3}\cos(2\theta)+\dots \\
&\quad + \alpha_{(\frac{p_3+e_3+2}{2}+1)p_3e_3}\cos((p_3+e_3+2)\theta)\Big)
 +a_{01,1}r^2\Big(\hat{\alpha}_{101}\sin(\theta)
 +\hat{\alpha}_{201}\sin(3\theta)\Big) \\
&\quad +\dots +a_{p_3q_3,1}r^{p_3+q_3+1}\Big(\hat{\alpha}_{1p_3q_3}
 \sin(\theta)+\hat{\alpha}_{2p_3q_3}\sin(3\theta)
 +\hat{\alpha}_{3p_3q_3}\sin(5\theta)+\dots \\
&\quad +\hat{\alpha}_{(\frac{p_3+q_3+3}{2})p_3q_3}
\sin((p_3+q_3+2)\theta)\Big)  +a_{10,1}r^2
 \Big(\hat{\alpha}_{110}+\hat{\alpha}_{210}\cos(\theta)\\
&\quad +\hat{\alpha}_{310}\cos(3\theta)\Big)+\dots
 +a_{c_3e_3,1}r^{c_3+e_3+1}\Big(\hat{\alpha}_{1c_3e_3}
+\hat{\alpha}_{2c_3e_3}\cos(\theta)+\hat{\alpha}_{3c_3e_3}\cos(3\theta)\\
&\quad +\dots  +\hat{\alpha}_{(\frac{c_3+e_3+1}{2}+2)c_3e_3}
 \cos((c_3+e_3+2)\theta)\Big)+a_{11,1}r^3\Big(\hat{\alpha}_{111}\theta
 +\hat{\alpha}_{211}\sin(4\theta)\Big) \\
&\quad +a_{13,1}r^5
\Big(\hat{\alpha}_{113}\theta+\hat{\alpha}_{213}\sin(2\theta)
 +\hat{\alpha}_{313}\sin(4\theta)+\hat{\alpha}_{413}\sin(6\theta)\Big)+\dots\\
&\quad  +a_{c_3q_3,1}r^{c_3+q_3+1}\Big(\hat{\alpha}_{1c_3q_3}\theta
 +\hat{\alpha}_{2c_3q_3}\sin(2\theta)\\
&\quad +\hat{\alpha}_{3c_3q_3}\sin(4\theta)+\dots
+\hat{\alpha}_{(c_3+q_3)c_3q_3}\sin((c_3+q_3+2)\theta)\Big).
\end{align*}
We know from \eqref{e3.3)} that $F_{10}$ is identically zero if
and only if $a_{ij,2}=0$ for all $i$ even, $j$ even.

Now using the integrals given at the appendix, we calculate
% \label{e4.2}
\begin{align*}
&H_1(r)\\
&=\frac{1}{2\pi} \int_0^{2\pi} \Big[
\frac{d}{dr}F_1(r,\theta)y_1(r,\theta)\Big]d\theta \\
&=\frac{1}{2}\Big[ \sum_{\substack{i+j=1\\ \text{$i$ even, $J$ odd}}}^n
(i+j+1)a_{ij,2}r^{i+j}\Big[a_{10,2}r^2\Big(
\alpha_{110}A_{ij}^1+\alpha_{210}A_{ij}^3\Big)+\dots \\
&\quad +a_{c_1e_1,2}r^{c_1+e_1+1}\Big(\alpha_{1c_1e_1}A_{ij}^1
+\alpha_{2c_1e_1}A_{ij}^3+\dots
+\alpha_{\frac{(c_1+e_1+2)+1}{2}c_1e_1}A_{ij}^{c_1+e_1+2}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=1\\ \text{$i$ odd, $j$ even}}}^n
 (i+j+1)a_{ij,2}r^{i+j}\Big[ a_{01,2}r^2 \Big(\alpha_{201}B_{ij}^1
 +\alpha_{301}B_{ij}^3\Big)+\dots \\
&\quad +a_{p_1q_1,2}r^{p_1+q_1+1} \Big(\alpha_{2p_1q_1}B_{ij}^1
 +\alpha_{3cp_1q_1}B_{ij}^3+\dots 
 + \alpha_{\frac{(p_1+q_1+2)+3}{2}p_1q_1}B_{ij}^{p_1+q_1+2}\Big) \Big]\\
&\quad + \sum_{\substack{i+j=1\\ \text{$i$ odd, $j$ even}}}^n
(i+j+1)a_{ij,2}r^{i+j}\Big[ b_{00,2}\Big(\tilde{\alpha}_{200}\tilde{B}_{ij}^1\Big)
+b_{02,2}r^2 \Big(\tilde{\alpha}_{202}\tilde{B}_{ij}^1
+\tilde{\alpha}_{302}\tilde{B}_{ij}^3\Big) \\
&\quad +\dots  +b_{p_2e_2,2}r^{p_2+e_2}\Big(\tilde{\alpha}_{2p_2e_2}\tilde{B}_{ij}^1
+\tilde{\alpha}_{302}\tilde{B}_{ij}^3+\dots
  +\tilde{\alpha}_{(\frac{p_2+e_2}{2}+2)p_2e_2}
\tilde{B}_{ij}^{p_2+e_2+1} \Big)\Big]\\
&\quad + \sum_{\substack{i+j=2 \\ \text{$i$ odd, $j$ odd}}} ^n
(i+j+1) a_{ij,2}r^{i+j}\Big[ b_{01,2}r\Big(\tilde{\alpha}_{101}
 \gamma_{ij}+\tilde{\alpha}_{201}C_{ij}^2\Big)+\dots \\
&\quad +b_{p_2q_2,2}r^{p_2+q_2}\Big(\tilde{\alpha}_{1p_2q_2}\gamma_{ij}
+\tilde{\alpha}_{2p_2q_2}C_{ij}^2
+\tilde{\alpha}_{3p_2q_2}C_{ij}^{4}+\dots  \\
&\quad +\tilde{\alpha}_{(\frac{p_2+q_2+3}{2})p_2q_2}C_{ij}^{p_2+q_2+1}\Big)\Big]\\
&\quad +\sum_{\substack{i+j=1 \\ \text{$i$ even, $j$ odd}}}^n
(i+j+1)a_{ij,2}r^{i+j}\Big[b_{11,2}r^2\Big( \tilde{\alpha}_{111}\tilde{A}_{ij}^1+
\tilde{\alpha}_{211}\tilde{A}_{ij}^3 \Big)+\dots \\
&\quad +b_{c_2q_2,2}r^{c_2+q_2}\Big(\tilde{\alpha}_{1c_2q_2}\tilde{A}_{ij}^1+
\tilde{\alpha}_{2c_2q_2}\tilde{A}_{ij}^3+\dots 
+\tilde{\alpha}_{(\frac{c_2+q_2+2}{2})c_2q_2}\tilde{A}_{ij}^{c_2+q_2+1}
 \Big)\Big]\\
&\quad + \sum_{\substack{i+j=1 \\ \text{$i$ even, $j$ odd}}}^n
(i+j+1)a_{ij,2}r^{i+j}\Big[a_{01,1}r^2\Big(\hat{\alpha}_{101}\hat{A}_{ij}^1+
\hat{\alpha}_{201}\hat{A}_{ij}^3\Big)\\
&\quad +\dots +a_{p_3q_3,1}r^{p_3+q_3+1}
\Big(\hat{\alpha}_{1p_3q_3}\hat{A}_{ij}^1
+ \hat{\alpha}_{2p_3q_3}\hat{A}_{ij}^3+\dots
+\hat{\alpha}_{(\frac{p_3+q_3+3}{2})p_3q_3}\hat{A}_{ij}^{p_3+q_3+2}\Big)
 \Big]\\
&\quad \sum_{\substack{i+j=1\\ \text{$i$ odd, $j$ even}}}^n
(i+j+1)a_{ij,2}r^{i+j}\Big[a_{10,1}r^2
\Big(\hat{\alpha}_{210}\hat{B}_{ij}^1+\hat{\alpha}_{310}\hat{B}_{ij}^3\Big)+\dots\\
&\quad +a_{c_3e_3,1}r^{c_3+e_3+1}\Big(\hat{\alpha}_{2c_3e_3}
\hat{B}_{ij}^1+\hat{\alpha}_{3c_3e_3}\hat{B}_{ij}^3+\dots 
 + \hat{\alpha}_{(\frac{c_3+e_3+1}{2}+2)c_3e_3}\hat{B}_{ij}^{c_3+e_3+2}\Big)
\Big]\\
&\quad + \sum_{\substack{i+j=2 \\ \text{$i$ odd, $j$ odd}}}^n
(i+j+1)a_{ij,2}r^{i+j}\Big[a_{11,1}r^3\Big(
\hat{\alpha}_{111}\gamma_{ij}+\hat{\alpha}_{211}\tilde{C}_{ij}^{4}\Big)+\dots \\
&\quad +a_{c_3q_3,1}r^{c_3+q_3+1}\Big(\hat{\alpha}_{1c_3q_3}\gamma_{ij}
+\hat{\alpha}_{2c_3q_3}\tilde{C}_{ij}^2+\dots 
 +\hat{\alpha}_{(c_3+q_3)c_3q_3}\tilde{C}_{ij}^{c_3+q_3+2}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=2\\ \text{$i$ even, $j$ even}}} ^m
(i+j)b_{ij,2}r^{i+j-1}\Big[ a_{10,2}r^2
\Big( \alpha_{110}D_{ij}^1+\alpha_{210}D_{ij}^3\Big)+\dots \\
&\quad +a_{c_1e_1,2}r^{c_1+e_1+1}\Big(
\alpha_{1c_1e_1}D_{ij}^1+\alpha_{2c_1e_1}D_{ij}^3+\dots 
 +\alpha_{\frac{(c_1+e_1+2)+1}{2}c_1e_1}D_{ij}^{c_1+e_1+2}\Big)\Big]\\
&\quad +\sum_{\substack{i+j=2\\ \text{$i$ odd, $j$ odd}}} ^m
(i+j)b_{ij,2}r^{i+j-1}\Big[a_{01,2}r^2\Big(\alpha_{201}E_{ij}^1
+\alpha_{301}E_{ij}^3\Big)+\dots \\
&\quad +a_{p_1q_1,2}r^{p_1+q_1+1}\Big(\alpha_{2p_1q_1}E_{ij}^1
+\alpha_{3p_1q_1}E_{ij}^3+\dots \\
&\quad + \alpha_{\frac{(p_1+q_1+2)+3}{2}p_1q_1}E_{ij}^{p_1+q_1+2}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^m
(i+j)b_{ij,2}r^{i+j-1}\Big[a_{11,2}r^3\Big(\alpha_{111}\beta_{ij}
+\alpha_{211}F_{ij}^2+\alpha_{311}F_{ij}^{4}\Big)+\dots \\
&\quad +a_{c_1q_1,2}r^{c_1+q_1+1}\Big(\alpha_{1c_1q_1}\beta_{ij}
 +\alpha_{2c_1q_1}F_{ij}^2+\alpha_{3c_1q_1}F_{ij}^{4}+\dots \\
&\quad +\alpha_{\frac{(c_1+q_1+2)+2}{2}c_1q_1}F_{ij}^{c_1+q_1+2}\Big)\Big]\\
&\quad +\sum_{\substack{i+j=2\\ \text{$i$ odd, $j$ odd}}}^m
(i+j)b_{ij,2}r^{i+j-1}\Big[b_{00,2}\Big(\tilde{\alpha}_{200}\tilde{E}_{ij}^1\Big)
+b_{02,2}r^2\Big(\tilde{\alpha}_{202}\tilde{E}_{ij}^1
+\tilde{\alpha}_{302}\tilde{E}_{ij}^3\Big)\\
&\quad +\dots  +b_{p_2e_2,2}r^{p_2+e_2}\Big(\tilde{\alpha}_{2p_2e_2}\tilde{E}_{ij}^1
+\tilde{\alpha}_{3p_2e_2}\tilde{E}_{ij}^3+\dots 
 + \tilde{\alpha}_{(\frac{p_2+e_2}{2}+2)p_2e_2}
 \tilde{E}_{ij}^{p_2+e_2+1}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=1 \\ \text{$i$ odd, $j$ even}}}^m
(i+j)b_{ij,2}r^{i+j-1}\Big[b_{01,2}r\Big(\tilde{\alpha}_{101}\sigma_{ij}
+\tilde{\alpha}_{201}G_{ij}^2\Big)+\dots \\
&\quad +b_{p_2q_2,2}r^{p_2+q_2}
\Big(\tilde{\alpha}_{1p_2q_21}\sigma_{ij}+\tilde{\alpha}_{2p_2q_2}G_{ij}^2+\dots \\
&\quad +\tilde{\alpha}_{(\frac{p_2+q_2+3}{2})p_2q_2}G_{ij}^{p_2+q_2+1}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^m
(i+j)b_{ij,2}r^{i+j-1}\Big[b_{10,2}r\Big(
\tilde{\alpha}_{110}\beta_{ij}+\tilde{\alpha}_{210}\tilde{F}_{ij}^2\Big)+\dots \\
&\quad + b_{c_2e_2,2}r^{c_2+e_2}\Big(\tilde{\alpha}_{1c_2e_2}\beta_{ij}
+\tilde{\alpha}_{210}\tilde{F}_{ij}^2+\dots
+\tilde{\alpha}_{(\frac{c_2+e_2+1}{2}+1)c_2e_2}\tilde{F}_{ij}^{c_2+e_2+1}
\Big)\Big]\\
&\quad + \sum_{\substack{i+j=2 \\\text{$i$ even, $j$ even}}}^m
(i+j)b_{ij,2}r^{i+j-1}\Big[b_{11,2}r^2
\Big( \tilde{\alpha}_{111}\tilde{D}_{ij}^1
+\tilde{\alpha}_{211}\tilde{D}_{ij}^3\Big) +\dots \\
&\quad +b_{c_2q_2,2}r^{c_2+q_2}\Big(\tilde{\alpha}_{1c_2q_2}\tilde{D}_{ij}^1+
\tilde{\alpha}_{2c_2q_2}\tilde{D}_{ij}^3+\dots
 +\tilde{\alpha}_{(\frac{c_2+q_2+2}{2})c_2q_2}\tilde{D}_{ij}^{c_2+q_2+1}
\Big)\Big]\\
&\quad+ \sum_{\substack{i+j=1 \\ \text{$i$ even, $j$ odd}}}^m
(i+j)b_{ij,2}r^{i+j-1}\Big[a_{00,1}r\Big(\hat{\alpha}_{100}
\beta_{ij}+\hat{\alpha}_{200}\hat{F}_{ij}^2\Big)+\dots \\
&\quad +a_{p_3e_3,1}r^{p_3+e_3+1}\Big(\hat{\alpha}_{1p_3e_3}\beta_{ij}
+\hat{\alpha}_{2p_3e_3}\hat{F}_{ij}^2+\dots 
 +\hat{\alpha}_{(\frac{p_3+e_3+2}{2}+1)p_3e_3}
 \hat{F}_{ij}^{p_3+e_3+2}\Big)\Big]\\
&\quad+ \sum_{\substack{i+j=2\\ \text{$i$ even, $j$ even}}}^m
(i+j)b_{ij,2}r^{i+j-1}\Big[a_{01,1}r^2\Big(\hat{\alpha}_{101}\hat{D}_{ij}^1
 +\hat{\alpha}_{201}\hat{D}_{ij}^3\Big)+\dots \\
&\quad +a_{p_3q_3,1}r^{p_3+q_3+1}\Big(\hat{\alpha}_{1p_3q_3}\hat{D}_{ij}^1+
\hat{\alpha}_{2p_3q_3}\hat{D}_{ij}^3+\dots 
  + \hat{\alpha}_{(\frac{p_3+q_3+3}{2})p_3q_3}
\hat{D}_{ij}^{p_3+q_3+2}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=2 \\ \text{$i$ odd, $j$ odd}}}^m
(i+j)b_{ij,2}r^{i+j-1}\Big[a_{10,1}r^2\Big(\hat{\alpha}_{210}
\hat{E}_{ij}^1+\hat{\alpha}_{310}\hat{E}_{ij}^3\Big)+\dots \\
&\quad +a_{c_3e_3,1}r^{c_3+e_3+1}\Big(\hat{\alpha}_{2c_3e_3}\hat{E}_{ij}^1
+\hat{\alpha}_{3c_3e_3}\hat{E}_{ij}^3+\dots 
+ \hat{\alpha}_{(\frac{c_3+e_3+1}{2}+2)c_3e_3}\hat{E}_{ij}^{c_3+e_3+2}\Big)
\Big]\\
&\quad + \sum_{\substack{i+j=1 \\ \text{$i$ odd, $j$ even}}}^m
(i+j)b_{ij,2}r^{i+j-1}\Big[a_{11,1}r^3\Big(\hat{\alpha}_{111}\sigma_{ij}
+\hat{\alpha}_{211}\tilde{G}_{ij}^{4}\Big)+\dots \\
&\quad + a_{c_3q_3,1}r^{c_3+q_3+1}\Big(\hat{\alpha}_{1c_3q_3}\sigma_{ij}
 +\hat{\alpha}_{2c_3q_3}\tilde{G}_{ij}^2
 + \hat{\alpha}_{3c_3q_3}\tilde{G}_{ij}^{4}+\dots\\
&\quad +\hat{\alpha}_{(c_3+q_3)c_3q_3}\tilde{G}_{ij}^{c_3+q_3+2}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=1 \\ \text{$i$ odd, $j$ even}}}^l
(i+j+1)a_{ij,1}r^{i+j}\Big[a_{10,2}r^2\Big(\alpha_{110}H_{ij}^1
+\alpha_{210}H_{ij}^3\Big)+\dots \\
&\quad +a_{c_1e_1,2}r^{c_1+e_1+1}\Big(\alpha_{1c_1e_1}H_{ij}^1
+\alpha_{2c_1e_1}H_{ij}^3+\dots 
 +\alpha_{\frac{(c_1+e_1+2)+1}{2}c_1e_1}H_{ij}^{c_1+e_1+2}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=1 \\ \text{$i$ even, $j$ odd}}}^l
(i+j+1)a_{ij,1}r^{i+j}\Big[a_{01,2}r^2\Big(\alpha_{201}I_{ij}^1
+\alpha_{301}I_{ij}^3\Big) +\dots \\
&\quad +a_{p_1q_1,2}r^{p_1+q_1+1}\Big(\alpha_{2p_1q_1}I_{ij}^1
+\alpha_{3p_1q_1}I_{ij}^3+\dots 
 +\alpha_{\frac{(p_1+q_1+2)+3}{2}p_1q_1}I_{ij}^{p_1+q_1+2}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=2 \\ \text{$i$ odd, $j$ odd}}}^l
(i+j+1)a_{ij,1}r^{i+j}\Big[a_{11,2}r^3\Big(
\alpha_{111}\delta_{ij}+\alpha_{211}K_{ij}^2+\alpha_{311}K_{ij}^{4}\Big)+\dots \\
&\quad +a_{c_1q_1,2}r^{c_1+q_1+1}\Big(\alpha_{1c_1q_1}\delta_{ij}+
\alpha_{21c_1q_1}K_{ij}^2+\dots 
 +\alpha_{\frac{(c_1+q_1+2)+2}{2}c_1q_1}K_{ij}^{c_1+q_1+2}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^l
(i+j+1)a_{ij,1}r^{i+j}\Big[b_{00,2}\Big(\tilde{\alpha}_{200}\tilde{I}_{ij}^1\Big)
+b_{02,2}r^2\Big(\tilde{\alpha}_{202}\tilde{I}_{ij}^1
+\tilde{\alpha}_{302}\tilde{I}_{ij}^3\Big) \\
&\quad +\dots  +b_{p_2e_2,2}r^{p_2+e_2}\Big( \tilde{\alpha}_{2p_2e_2}\tilde{I}_{ij}^1+\dots
 +\tilde{\alpha}_{(\frac{p_2+e_2}{2}+2)p_2e_2}\tilde{I}_{ij}^{p_2+e_2+1}\Big)
\Big]\\
&\quad +\sum_{\substack{i+j=0\\ \text{$i$ even, $j$ even}}}^l
(i+j+1)a_{ij,1}r^{i+j}\Big[ b_{01,2}r\Big(\tilde{\alpha}_{101}\mu_{ij}
+\tilde{\alpha}_{201}L_{ij}^2\Big)+\dots \\
&\quad +b_{p_2q_2,2}r^{p_2+q_2}\Big(\tilde{\alpha}_{1p_2q_2}\mu_{ij}
+\tilde{\alpha}_{2p_2q_2}L_{ij}^2+\dots 
 + \tilde{\alpha}_{(\frac{p_2+q_2+3}{2})p_2q_2}L_{ij}^{p_2+q_2+1}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=2 \\ \text{$i$ odd, $j$ odd}}}^l
(i+j+1)a_{ij,1}r^{i+j}\Big[b_{10,2}r\Big(\tilde{\alpha}_{110}\delta_{ij}
+\tilde{\alpha}_{210}\tilde{K}_{ij}^2\Big)+\dots \\
&\quad + b_{c_2e_2,2}r^{c_2+e_2}\Big(\tilde{\alpha}_{1c_2e_2}\delta_{ij}+
\tilde{\alpha}_{2c_2e_2}\tilde{K}_{ij}^2+\dots 
 +\tilde{\alpha}_{(\frac{c_2+e_2+1}{2}+1)c_2e_2}\tilde{K}_{ij}^{c_2+e_2+1}
 \Big)\Big]\\
&\quad + \sum_{\substack{i+j=1 \\ \text{$i$ odd, $j$ even}}}^l
(i+j+1)a_{ij,1}r^{i+j}\Big[b_{11,2}r^2\Big( \tilde{\alpha}_{111}\tilde{H}_{ij}^1+
\tilde{\alpha}_{211}\tilde{H}_{ij}^3\Big)+\dots \\
&\quad +b_{c_2q_2,2}r^{c_2+q_2} \Big( \tilde{\alpha}_{1c_2q_2}\tilde{H}_{ij}^1+\dots
 +\tilde{\alpha}_{(\frac{c_2+q_2+2}{2})c_2q_2}\tilde{H}_{ij}^{c_2+q_2+1}\Big)
 \Big]\\
&\quad +\sum_{\substack{i+j=2\\ \text{$i$ odd, $j$ odd}}}^l
(i+j+1)a_{ij,1}r^{i+j}\Big[a_{00,1}r\Big(
 \hat{\alpha}_{100}\delta_{ij}+ \hat{\alpha}_{200}\hat{K}_{ij}^2\Big) +\dots\\
&\quad + a_{p_3e_3,1}r^{p_3+e_3+1}\Big(\hat{\alpha}_{1p_3e_3}\delta_{ij}+
\hat{\alpha}_{2p_3e_3}\hat{K}_{ij}^2+\dots 
 + \hat{\alpha}_{(\frac{p_3+e_3+2}{2}+1)p_3e_3}
 \hat{K}_{ij}^{p_3+e_3+2}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=1\\ \text{$i$ odd, $j$ even}}}^l
(i+j+1)a_{ij,1}r^{i+j}\Big[a_{01,1}r^2\Big(
 \hat{\alpha}_{101}\hat{H}_{ij}^1+ \hat{\alpha}_{201}\hat{H}_{ij}^3\Big) +\dots \\
&\quad + a_{p_3q_3,1}r^{p_3+q_3+1}\Big(
\hat{\alpha}_{1p_3q_3}\hat{H}_{ij}^1+
 \hat{\alpha}_{2p_3q_3}\hat{H}_{ij}^3+\dots 
 + \hat{\alpha}_{(\frac{p_3+q_3+3}{2})p_3q_3}\hat{H}_{ij}^{p_3+q_3+2}\Big)\Big]\\
&\quad + \sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^l
(i+j+1)a_{ij,1}r^{i+j}\Big[a_{10,1}r^2\Big(
 \hat{\alpha}_{210}\hat{I}_{ij}^1+ \hat{\alpha}_{310}\hat{I}_{ij}^3\Big) +\dots \\
&\quad +a_{c_3e_3,1}r^{c_3+e_3+1} \Big(
\hat{\alpha}_{2c_3e_3}\hat{I}_{ij}^1+\dots 
 + \hat{\alpha}_{(\frac{c_3+e_3+1}{2}+1)c_3e_3}\hat{I}_{ij}^{c_3+e_3+2}
\Big)\Big]\\ 
&\quad + \sum_{\substack{i+j=0\\ \text{$i$ even, $j$ even}}}^l
(i+j+1)a_{ij,1}r^{i+j}\Big[a_{11,1}r^3\Big( \hat{\alpha}_{111}\mu_{ij}
+ \hat{\alpha}_{211}\tilde{L}_{ij}^{4}\Big) +\dots \\
&\quad +a_{c_3q_3,1} r^{c_3+q_3+1} \Big(
\hat{\alpha}_{1c_3q_3}\mu_{ij}+
\hat{\alpha}_{2c_3q_3}\tilde{L}_{ij}^2+\dots 
+ \hat{\alpha}_{(c_3+q_3)c_3q_3}\tilde{L}_{ij}^{c_3+q_3+2}
\Big)\Big]\Big].
\end{align*}

We also note that 
\begin{align*}
&F_2(r,\theta)\\
&=\sum_{i+j=0}^nc_{ij,2}r^{i+j+1}\cos^{i}(\theta)\sin^{j+2}(\theta)+
\sum_{i+j=0}^m d_{ij,2}r^{i+j}\cos^{i}(\theta)\sin^{j+1}(\theta) \\
&\quad +\sum_{i+j=0}^lC_{ij,1}r^{i+j+1}\cos^{i+1}(\theta)\sin^{j+1}(\theta)\\
&\quad - \sum_{\substack{i+j=0 \\ \text{$i$ odd  or $j$  odd}}}^n
\sum_{\substack{k+h=0\\ \text{$k$ odd  or $h$ odd}}}^n
a_{ij,2}a_{kh,2}r^{i+j+k+h+1} \cos^{i+k+1}(\theta)\sin^{j+h+3}(\theta)\\
&\quad -\sum_{\substack{i+j=0 \\ \text{$i$ odd  or $j$ odd}}}^n
\sum_{k+h=0}^m a_{ij,2}b_{kh,2}r^{i+j+k+h}
\cos^{i+k+1}(\theta)\sin^{j+h+2}(\theta)\\
&\quad + \sum_{\substack{i+j=0 \\ \text{$i$ odd  or $j$ odd}}}^n
\sum_{k+h=0}^la_{ij,2}a_{kh,1}r^{i+j+k+h+1}
\cos^{i+k}(\theta)\sin^{j+h+4}(\theta)\\
&\quad -\sum_{i+j=0}^m \sum_{\substack{k+h=0 \\ \text{$k$ odd  or $h$ odd}}}^n
b_{ij,2}a_{kh,2}r^{i+j+k+h}
\cos^{i+k+1}(\theta)\sin^{j+h+2}(\theta)\\
&\quad -\sum_{i+j=0}^m \sum_{k+h=0}^m b_{ij,2}b_{kh,2}r^{i+j+k+h-1}
\cos^{i+k+1}(\theta)\sin^{j+h+1}(\theta)\\
&\quad +\sum_{i+j=0}^m \sum_{k+h=0}^lb_{ij,2}a_{kh,1}r^{i+j+k+h}
\cos^{i+k}(\theta)\sin^{j+h+3}(\theta)\\
&\quad -\sum_{\substack{i+j=0\\ \text{$k$ odd or $h$ odd}}}^l
\sum_{k+h=0}^n a_{ij,1}a_{kh,2}r^{i+j+k+h+1} \cos^{i+k+2}(\theta)\sin^{j+h+2}(\theta)\\
&\quad -\sum_{i+j=0}^l\sum_{k+h=0}^m a_{ij,1}b_{kh,2}r^{i+j+k+h}
\cos^{i+k+2}(\theta)\sin^{j+h+1}(\theta)\\
&\quad +\sum_{i+j=0}^l\sum_{k+h=0}^la_{ij,1}a_{kh,1}r^{i+j+k+h+1}
\cos^{i+k+1}(\theta)\sin^{j+h+3}(\theta).
\end{align*}
Again, using the integrals given in the appendix, we  calculate
\begin{align*}
H_2(r)&=\frac{1}{2\pi}\int_0^{2\pi}F_2(r,\theta)d\theta \\
&=\frac{1}{2}\Big[\sum_{\substack{i+j=0 \\ \text{$i$ even, $j$ even}}}^n
c_{ij,2}r^{i+j+1}\alpha_{ij}
+\sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^m
 d_{ij,2}r^{i+j}\beta_{ij}\\
&\quad + \sum_{\substack{i+j=2\\ \text{$i$ odd, $j$ odd}}}^l
C_{ij,1}r^{i+j+1}\delta_{ij}\\
&\quad - \sum_{\substack{i+j=1\\ \text{$i$ odd, $j$ even}}}^n
\sum_{\substack{k+h=1\\ \text{$k$ even, $h$ odd}}}^n
 a_{ij,2}a_{kh,2}r^{i+j+k+h+1}M_{ijkh} \\
&\quad- \sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^n
 \sum_{\substack{k+h=1 \\ \text{$k$ odd, $h$ even}}}^n
 a_{ij,2}a_{kh,2}r^{i+j+k+h+1}\tilde{M}_{ijkh}\\
&\quad-\sum_{\substack{i+j=1\\ \text{$i$ odd, $j$ even}}}^n
\sum_{\substack{k+h=0\\ \text{$k$ even, $h$ even}}}^m
a_{ij,2}b_{kh,2}r^{i+j+k+h}N_{ijkh}\\
&\quad -\sum_{\substack{i+j=2\\ \text{$i$ odd, $j$ odd}}}^n
 \sum_{\substack{k+h=1\text{$k$ even, $h$ odd}}}^m
a_{ij,2}b_{kh,2}r^{i+j+k+h}\tilde{N}_{ijkh}\\
&\quad -\sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^n
 \sum_{\substack{k+h=2\\ \text{$k$ odd, $h$ odd}}}^m
 a_{ij,2}b_{kh,2}r^{i+j+k+h}\hat{N}_{ijkh}\\
&\quad +\sum_{\substack{i+j=1\\ \text{$i$ odd, $j$ even}}}^n
\sum_{\substack{k+h=1\\ \text{$k$ odd, $h$ even}}}^l
a_{ij,2}a_{kh,1}r^{i+j+k+h+1}P_{ijkh}\\
&\quad +\sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^n
\sum_{\substack{k+h=1\\ \text{$k$ even, $h$ odd}}}^l
 a_{ij,2}a_{kh,1}r^{i+j+k+h+1}\tilde{P}_{ijkh}\\
&\quad +\sum_{\substack{i+j=2\\ \text{$i$ odd, $j$ odd}}}^n
\sum_{\substack{k+h=2\\ \text{$k$ odd, $h$ odd}}}^l
 a_{ij,2}a_{kh,1}r^{i+j+k+h+1}\hat{P}_{ijkh}\\
&\quad - \sum_{\substack{i+j=2\\ \text{$i$ odd, $j$ odd}}}^m
\sum_{\substack{k+h=1\\  \text{$k$ even, $h$ odd}}}^n
b_{ij,2}a_{kh,2}r^{i+j+k+h}\tilde{N}_{ijkh}\\
&\quad- \sum_{\substack{i+j=0\\ \text{$i$ even, $j$ even}}}^m
\sum_{\substack{k+h=1 \\  \text{$k$ odd, $h$  even}}}^n
b_{ij,2}a_{kh,2}r^{i+j+k+h}N_{ijkh}\\
&\quad -\sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^m
\sum_{\substack{k+h=2\\  \text{$k$ odd, $h$  odd}}}^n
 b_{ij,2}a_{kh,2}r^{i+j+k+h}\hat{N}_{ijkh}\\
&\quad -\sum_{\substack{i+j=0\\ \text{$i$ even, $j$ even}}}^m
\sum_{\substack{k+h=2\\  \text{$k$ odd, $h$  odd}}}^m
 b_{ij,2}b_{kh,2}r^{i+j+k+h-1}Q_{ijkh}\\
&\quad -\sum_{\substack{i+j=2\\ \text{$i$ odd, $j$  odd}}}^m
\sum_{\substack{k+h=0\\  \text{$k$ even, $h$ even}}}^m
 b_{ij,2}b_{kh,2}r^{i+j+k+h-1}Q_{ijkh}\\
&\quad -\sum_{\substack{i+j=1\\ \text{$i$ odd, $j$ even}}}^m
\sum_{\substack{k+h=1\\  \text{$k$ even, $h$ odd}}}^m
b_{ij,2}b_{kh,2}r^{i+j+k+h-1}\tilde{Q}_{ijkh}\\
&\quad -\sum_{\substack{i+j=1\\\text{$i$ even, $j$ odd}}}^m
\sum_{\substack{k+h=1 \\ \text{$k$ odd, $h$  even}}}^m
 b_{ij,2}b_{kh,2}r^{i+j+k+h-1}\tilde{Q}_{ijkh}\\
&\quad +\sum_{\substack{i+j=0\\ \text{$i$ even, $j$ even}}}^m
 \sum_{\substack{k+h=1\\ \text{$k$ even, $h$ odd}}}^l
 b_{ij,2}a_{kh,1}r^{i+j+k+h}R_{ijkh}\\
&\quad +\sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^m
\sum_{\substack{k+h=0\\  \text{$k$ even, $h$ even}}}^l
 b_{ij,2}a_{kh,1}r^{i+j+k+h}R_{ijkh}\\
&\quad +\sum_{\substack{i+j=1\\ \text{$i$ odd, $j$  even}}}^m
\sum_{\substack{k+h=2 \\  \text{$k$ odd, $h$  odd}}}^l
 b_{ij,2}a_{kh,1}r^{i+j+k+h}\tilde{R}_{ijkh}\\
&\quad + \sum_{\substack{i+j=2\\ \text{$i$ odd, $j$  odd}}}^m
\sum_{\substack{k+h=1\\  \text{$k$ odd, $h$  even}}}^l
 b_{ij,2}a_{kh,1}r^{i+j+k+h}\tilde{R}_{ijkh}\\
&\quad - \sum_{\substack{i+j=1// \text{$i$ even, $j$ odd}}}^l
\sum_{\substack{k+h=1 \\  \text{$k$ even, $h$ odd}}}^n
 a_{ij,1}a_{kh,2}r^{i+j+k+h+1}T_{ijkh}\\
&\quad-\sum_{\substack{i+j=1\\ \text{$i$ odd, $j$ even}}}^l
\sum_{\substack{k+h=1\\  \text{$k$ odd, $h$  even}}}^n
 a_{ij,1}a_{kh,2}r^{i+j+k+h+1}\tilde{T}_{ijkh}\\
&\quad -\sum_{\substack{i+j=2\\ \text{$i$ odd, $j$  odd}}}^l
\sum_{\substack{k+h=2 \\  \text{$k$ odd, $h$  odd}}}^n
 a_{ij,1}a_{kh,2}r^{i+j+k+h+1}\hat{T}_{ijkh}\\
&\quad - \sum_{\substack{i+j=0\\ \text{$i$ even, $j$ even}}}^l
\sum_{\substack{k+h=1\\  \text{$k$ even, $h$ odd}}}^m
 a_{ij,1}b_{kh,2}r^{i+j+k+h}U_{ijkh}\\
&\quad- \sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^l
\sum_{\substack{k+h=0\\  \text{$k$ even, $h$ even}}}^m
 a_{ij,1}b_{kh,2}r^{i+j+k+h}U_{ijkh}\\
&\quad- \sum_{\substack{i+j=1\\ \text{$i$ odd, $j$ even}}}^l
\sum_{\substack{k+h=2\\ \text{$k$ odd, $h$  odd}}}^m
 a_{ij,1}b_{kh,2}r^{i+j+k+h}\tilde{U}_{ijkh}\\
&\quad-\sum_{\substack{i+j=2 \\ \text{$i$ odd, $j$  odd}}}^l
\sum_{\substack{k+h=1\\  \text{$k$ odd, $h$ even}}}^m
 a_{ij,1}b_{kh,2}r^{i+j+k+h}\tilde{U}_{ijkh}\\
 &\quad +\sum_{\substack{i+j=0\\ \text{$i$ even, $j$  even}}}^l
\sum_{\substack{k+h=2\\  \text{$k$ odd, $h$  odd}}}^l
 a_{ij,1}a_{kh,1}r^{i+j+k+h+1}V_{ijkh}\\
&\quad +\sum_{\substack{i+j=1\\ \text{$i$ even, $j$ odd}}}^l
\sum_{\substack{k+h=1\\  \text{$k$ odd, $h$  even}}}^l
a_{ij,1}a_{kh,1}r^{i+j+k+h+1}\tilde{V}_{ijkh}\\
&\quad + \sum_{\substack{i+j=1\\ \text{$i$ odd, $j$ even}}}^l
\sum_{\substack{k+h=1 \\  \text{$k$ even, $h$  odd}}}^l
a_{ij,1}a_{kh,1}r^{i+j+k+h+1}\tilde{V}_{ijkh}\Big].
\end{align*}

Finally, we obtain  $H_1(r)+H_2(r)$ is a polynomial in the
variable $r^2$ of the form
$$
H_1(r)+H_2(r)=r\Big[P_1(r^2)+r^2 P_2(r^2)+r^4 P_3(r^2)\Big],
$$
where
$P_1(r^2)$ is a polynomial in the variable $r^2$ of degree
\begin{align*}
\lambda_1&=\max \Big\{[\frac{O(m)+E(l)-1}{2}], [\frac{E(m)+O(l)-1}{2}],
[\frac{O(n)+O(l)-2}{2}], \\
&\quad m-1,[\frac{E(n)}{2}] , [\frac{E(m)+O(n)-1}{2}]\Big\},
\end{align*}
$P_2(r^2)$ is a polynomial in the variable $r^2$ of degree
\begin{align*}
\lambda_2
&=\max\Big\{O(n)-1, [\frac{E(m)+O(n)-3}{2}], [\frac{O(m)+E(n)-3}{2}],
[\frac{O(n)+O(l)-2}{2}], \\
&\quad l-1,  E(m)-2, [\frac{E(m)+O(n)-3}{2}], [\frac{O(m)+E(l)-3}{2}],
  [\frac{E(n)+O(m)-3}{2}] \Big\},
\end{align*}
$P_3(r^2)$ is a polynomial in the variable $r^2$ of degree
$$
\lambda_3=[\frac{E(n)+E(l)-4}{2}],
$$
where $O(i)$ is the largest odd integer less than or equal to $i$,
$E(i)$ is the largest even integer less than or
equal to $i$ and $[\cdot]$ denotes the integer part function.
Then
$$
F_{20}=\frac{1}{2\pi}r\Big[P_1(r^2)+r^2 P_2(r^2)+r^4
P_3(r^2)\Big].
$$
To find the real positive roots of $F_{20}$ we must find the zeros
of a polynomial in $r^2$ of degree $\lambda=\max \{\lambda_1,
\lambda_2+1, \lambda_3+2 \}.$ This yields that $F_{20}$ has at
most $\lambda$ real positive roots. Hence, the Theorem \ref{thm1.2} is proved.

Moreover, we can choose the
coefficients $a_{ij,1}, a_{ij,2}, b_{ij,2}, C_{ij,1}, c_{ij,2},
d_{ij,2}$ in such a way that $F_{20}$ has exactly $\lambda$ real
positive roots.
In fact, we consider the example
\begin{equation}\label{exp5}
\begin{gathered}
\dot{x}= y-\varepsilon(1+xy-2x^2-x)y-\varepsilon^2(-2xy-y^2)y,\\
\dot{y}=-x-\varepsilon(4xy+(-x^2-y)y)-\varepsilon^2(x^2-y+(-xy+y^2-x)y),
\end{gathered}
\end{equation}
in polar coordinates. System \eqref{exp5} becomes
\begin{gather*}
\begin{aligned}
\dot{r}
&= \varepsilon\Big[-\cos(\theta)\sin(\theta)r+(\sin(\theta)
 -4\sin^2(\theta)\cos(\theta))r^2+2\sin(\theta)\cos^3(\theta)r^3\Big]\\
&\quad +\varepsilon^2\Big[\sin^2(\theta)r+(-\sin(\theta)\cos^2(\theta)
 +\sin^2(\theta)\cos(\theta))r^2+(2\cos(\theta)\sin(\theta) \\
&\quad +3\cos^2(\theta)\sin^2(\theta)
 -2\cos^3-(\theta)\sin(\theta)-\sin^2(\theta))r^3\Big],
\end{aligned} \\
\begin{aligned}
\dot{\theta}
&= -1+\varepsilon\Big[1-\cos^2(\theta)-4\sin(\theta)\cos^2r
 +(\cos(\theta)\sin(\theta)+2\cos^{4}(\theta)\\
&\quad -2\cos^2(\theta))r^2\Big]
 +\varepsilon^2\Big[\cos(\theta)\sin(\theta)
 +(-\cos^3(\theta)+\sin(\theta)\cos^2(\theta))r \\
&\quad +(3\sin(\theta)\cos^3(\theta)-3\cos(\theta)\sin(\theta)
 -2\cos^{4}(\theta)+3\cos^2(\theta)-1)r^2\Big].
\end{aligned}
\end{gather*}
Taking $\theta$ as the new independent variable, this system
becomes
$$
\frac{dr}{d\theta}=\varepsilon F_1(r,\theta)
+\varepsilon ^2 F_2(r,\theta)+O(\varepsilon^3),
$$
where
\[
F_1(r,\theta)=\cos(\theta)r\sin(\theta)-r^2\sin(\theta)
-2r^3\sin(\theta)\cos^3(\theta)+4r^2\sin^2(\theta)\cos(\theta),
\]
\begin{align*}
F_2(r,\theta)
&= \cos(s)r\sin(s)-r^2\sin(s)-\cos^3(s)r\sin(s)-4\sin(s)r^5\cos^7(s)\\
&\quad +4r^5\sin(s)\cos^5(s) -6r^4\sin(s)\cos^4(s)+6r^4\sin(s)\cos^2(s)\\
&\quad -18r^3\sin(s)\cos^3(s)-2\sin(s)r^3\cos(s)
 +20\cos^5(s)r^3\sin(s) \\
&\quad +2r^2\sin(s)\cos^2(s)  -r^4\sin^2(s)\cos(s)+16r^4\sin^2(s)\cos^5(s) \\
&\quad -2r^5\sin^2(s)\cos^4(s)  -8r^2\sin^2(s)\cos^3(s)+2r^3\sin^2(s)\cos^2(s)\\
&\quad +r^3\sin^2(s) -r\sin^2(s) 
 +3r^2\sin^2(s)\cos(s)-8r^4\sin^2(s)\cos^3(s).
\end{align*}
We have that $F_{10}(r)$ is identically zero, so to look for the limit cycles,
 we must solve the equation $F_{20}(r)=0$ which is equivalent to
$$
r(-\frac{1}{8}r^{4}-\frac{3}{4}r^3-\frac{1}{2})=0.
$$
This equation has exactly the two positive roots
$$
r_1=\frac{1}{2}\sqrt{10}+\frac{1}{2}\sqrt{2}, \quad
r_2=\frac{1}{2}\sqrt{10}-\frac{1}{2}\sqrt{2}.
$$
So system \eqref{exp5} has exactly two limit cycles bifurcating from the
periodic orbits of the linear center $\dot{x}=y$, $\dot{y}=-x$.


\section{Appendix}
In this appendix, we recall some formulae used during
this article; for more details see  \cite{AS}.
For $i\geq0$ and $j\geq0$, we have
\begin{gather*}
\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+2}(\theta)d\theta
= \begin{cases}
\pi\alpha_{ij}& \text{if $ i$ even, $j$ even,}\\
0 & \text{otherwise},
\end{cases} \\
\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+2}(\theta)\cos((2h)\theta)d\theta=0
\quad \text{if $i$ odd or $j$ odd},\;  h=0,1,2,\dots \\
\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+1}(\theta)d\theta
= \begin{cases}
\pi\beta_{ij}& \text{if $i$ even, $j$ odd,}\\
0 & \text{otherwise},
\end{cases} \\
\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+2}(\theta)\times \theta d\theta
= \begin{cases}
\pi\gamma_{ij}& \text{if $i$   odd, $j$ odd},\\
0 & \text{otherwise},
\end{cases} \\
\int_0^{2\pi}\cos^{i+1}(\theta)\sin^{j+1}(\theta)\times \theta d\theta
= \begin{cases}
\pi\mu_{ij}& \text{if $i$  even, $j$ even},\\
0 & \text{otherwise},
\end{cases} \\
\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+1}(\theta)\times \theta d\theta
= \begin{cases}
\pi\sigma_{ij}& \text{if $i$ odd, $j$  even},\\
0 & \text{otherwise},
\end{cases} \\
\int_0^{2\pi}\cos^{i+1}(\theta)\sin^{j+1}(\theta)d\theta
= \begin{cases}
\pi\delta_{ij}& \text{if $i$ odd, $j$  odd},\\
0 & \text{otherwise},
\end{cases} \\
\begin{aligned}
&\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+2}(\theta)\cos((2h+1)\theta)d\theta\\
&=\begin{cases}
\pi B_{ij}^{2h+1}& \text{if $i$ odd, $j$  even},\;  h=0,1,2,\dots ,n\\
\pi \tilde{B}_{ij}^{2h+1}& \text{if $i$ odd, $j$  even},\;  h=0,1,2,\dots ,m\\
\pi \hat{B}_{ij}^{2h+1}& \text{if $i$ odd, $j$  even},\; h=0,1,2,\dots ,l\\
 0 & \text{otherwise},
\end{cases}
\end{aligned}\\
\begin{aligned}
&\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+2}(\theta)\sin((2h+1)\theta)d\theta \\
&=\begin{cases}
\pi A_{ij}^{2h+1}& \text{ if $i$  even, $j$  odd},\; h=0,1,2,\dots ,n\\
\pi \tilde{A}_{ij}^{2h+1}& \text{if $i$  even, $j$  odd},\; h=0,1,2,\dots ,m\\
\pi \hat{A}_{ij}^{2h+1}& \text{if $i$  even, $j$  odd},\; h=0,1,2,\dots ,l\\
 0 & \text{otherwise},
\end{cases}
\end{aligned} \\
\begin{aligned}
&\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+2}(\theta)\sin((2h)\theta)d\theta \\
&=\begin{cases}
\pi C_{ij}^{2h}& \text{if $i$ odd, $j$  odd},\; h=0,1,2,\dots ,m\\
\pi \tilde{C}_{ij}^{2h}& \text{if $i$ odd, $j$  odd},\; h=0,1,2,\dots ,l\\
 0 & \text{otherwise},
\end{cases}
\end{aligned} \\
\begin{aligned}
&\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+1}(\theta)\sin((2h+1)\theta)d\theta \\
&=\begin{cases}
\pi D_{ij}^{2h+1}& \text{if $i$  even, $j$  even},\; h=0,1,2,\dots ,n\\
\pi \tilde{D}_{ij}^{2h+1}& \text{if $i$  even, $j$  even},\; h=0,1,2,\dots ,m\\
\pi \hat{D}_{ij}^{2h+1}& \text{if $i$  even, $j$  even},\; h=0,1,2,\dots ,l\\
 0 & \text{otherwise},
\end{cases}
\end{aligned}\\
\begin{aligned}
&\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+1}(\theta)\cos((2h+1)\theta)d\theta \\
&=\begin{cases}
\pi E_{ij}^{2h+1}& \text{if $i$ odd, $j$  odd},\; h=0,1,2,\dots ,n\\
\pi \tilde{E}_{ij}^{2h+1}& \text{if $i$ odd, $j$  odd},\; h=0,1,2,\dots ,m\\
\pi \hat{E}_{ij}^{2h+1}& \text{if $i$ odd, $j$  odd},\; h=0,1,2,\dots ,l\\
 0 & \text{otherwise},
\end{cases}
\end{aligned} \\
\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+1}(\theta)\cos((2h)\theta)d\theta 
=\begin{cases}
\pi F_{ij}^{2h}& \text{if $i$  even, $j$  odd},\; h=0,1,2,\dots ,n\\
\pi \tilde{F}_{ij}^{2h}& \text{if $i$  even, $j$  odd},\; h=0,1,2,\dots ,m\\
\pi \hat{F}_{ij}^{2h}& \text{if $i$  even, $j$  odd},\; h=0,1,2,\dots ,l\\
 0 & \text{otherwise},
\end{cases}
\\
\int_0^{2\pi}\cos^{i}(\theta)\sin^{j+1}(\theta)\sin((2h)\theta)d\theta 
=\begin{cases}
\pi G_{ij}^{2h}& \text{if $i$ odd, $j$  even},\; h=0,1,2,\dots ,m\\
\pi \tilde{G}_{ij}^{2h}& \text{if $i$ odd, $j$  even},\; h=0,1,2,\dots ,l\\
 0 & \text{otherwise},
\end{cases}
 \\ 
\begin{aligned}
&\int_0^{2\pi}\cos^{i+1}(\theta)\sin^{j+1}(\theta)\sin((2h+1)\theta)d\theta \\
&=\begin{cases}
\pi H_{ij}^{2h+1}& \text{if $i$ odd, $j$  even},\; h=0,1,2,\dots ,n\\
\pi \tilde{H}_{ij}^{2h+1}& \text{if $i$ odd, $j$  even},\; h=0,1,2,\dots ,m\\
\pi \hat{H}_{ij}^{2h+1}& \text{if $i$ odd, $j$  even},\; h=0,1,2,\dots ,l\\
 0 & \text{otherwise},
\end{cases}
\end{aligned}\\
\begin{aligned}
&\int_0^{2\pi}\cos^{i+1}(\theta)\sin^{j+1}(\theta)\cos((2h+1)\theta)d\theta\\
&=\begin{cases}
\pi I_{ij}^{2h+1}& \text{if $i$  even, $j$  odd},\; h=0,1,2,\dots ,n\\
\pi \tilde{I}_{ij}^{2h+1}& \text{if $i$  even, $j$  odd},\; h=0,1,2,\dots ,m\\
\pi \hat{I}_{ij}^{2h+1}& \text{if $i$  even, $j$  odd},\; h=0,1,2,\dots ,l\\
 0 & \text{otherwise},
\end{cases}
\end{aligned}\\
\int_0^{2\pi}\cos^{i+1}(\theta)\sin^{j+1}(\theta)\cos((2h)\theta)d\theta 
=\begin{cases}
\pi K_{ij}^{2h}& \text{if $i$ odd, $j$  odd},\; h=0,1,2,\dots ,n\\
\pi \tilde{K}_{ij}^{2h}& \text{if $i$ odd, $j$  odd},\; h=0,1,2,\dots ,m\\
\pi \hat{K}_{ij}^{2h}& \text{if $i$ odd, $j$  odd},\; h=0,1,2,\dots ,l\\
 0 & \text{otherwise},
\end{cases}
\\
\int_0^{2\pi}\cos^{i+1}(\theta)\sin^{j+1}(\theta)\sin((2h)\theta)d\theta
=\begin{cases}
\pi L_{ij}^{2h}& \text{if $i$  even, $j$  even},\; h=0,1,2,\dots ,m\\
\pi \tilde{L}_{ij}^{2h}& \text{if $i$  even, $j$  even},\; h=0,1,2,\dots ,l\\
 0 & \text{otherwise},
\end{cases}
\end{gather*}
where $\alpha_{ij}$, $\beta_{ij}$, $\gamma_{ij}, \delta_{ij}$,
$A_{ij}^{2h+1}$, $\tilde{A}_{ij}^{2h+1}$, $\hat{A}_{ij}^{2h+1}$,
$B_{ij}^{2h+1}$, $\tilde{B}_{ij}^{2h+1}$, $ \hat{B}_{ij}^{2h+1}$,
$C_{ij}^{2h}$, $\tilde{C}_{ij}^{2h}$, $D_{ij}^{2h+1}$,
$\tilde{D}_{ij}^{2h+1}$, $\hat{D}_{ij}^{2h+1}$, $E_{ij}^{2h+1}$,
$\tilde{E}_{ij}^{2h+1}$, $\hat{E}_{ij}^{2h+1}$, $F_{ij}^{2h+1}$,
$\tilde{F}_{ij}^{2h+1}$, $\hat{F}_{ij}^{2h+1}$, $ G_{ij}^{2h}$,
$\tilde{G}_{ij}^{2h}$, $H_{ij}^{2h+1}$, $\tilde{H}_{ij}^{2h+1}$,
$\hat{H}_{ij}^{2h+1}$, $I_{ij}^{2h+1}$, $\tilde{I}_{ij}^{2h+1}$,
$\hat{I}_{ij}^{2h+1}$, $K_{ij}^{2h}$, $\tilde{K}_{ij}^{2h}$,
$\hat{K}_{ij}^{2h}$, $L_{ij}^{2h}$ and
$\tilde{L}_{ij}^{2h} $ are non-zero constants.
\begin{gather*}
\begin{aligned}
&\int_0^{2\pi}\cos^{i+k+1}(\theta)\sin^{j+h+3}(\theta)d\theta\\
&=\begin{cases}
\pi M_{ijkh} & \text{if $i$ odd, $j$  even, $k$  even, $h$  odd},\\
\pi \tilde{M}_{ijkh} & \text{if $i$  even, $j$  odd, $k$  odd, $h$  even},\\
 0 & \text{otherwise},
\end{cases}
\end{aligned}\\
\begin{aligned}
&\int_0^{2\pi}\cos^{i+k+1}(\theta)\sin^{j+h+2}(\theta)d\theta \\
&=\begin{cases}
\pi N_{ijkh} & \text{if $i$ odd, $j$  even, $k$ even, $h$ even},\\
& \text{or $i$  even, $j$  even, $k$ odd, $h$  even},\\
\pi \tilde{N}_{ijkh} & \text{if $i$ odd, $j$  odd, $k$  even, $h$  is odd},\\
\pi \hat{N}_{ijkh} & \text{if $i$  even, $j$  odd, $k$  odd, $h$  odd},\\
 0 & \text{otherwise},
\end{cases}
\end{aligned}\\
\begin{aligned}
&\int_0^{2\pi}\cos^{i+k}(\theta)\sin^{j+h+4}(\theta)d\theta \\
&=\begin{cases}
\pi P_{ijkh} & \text{if $i$ odd, $j$  even, $k$ odd, $h$  even},\\
\pi \tilde{P}_{ijkh} & \text{if $i$  even, $j$  odd, $k$  even, $h$  odd},\\
\pi \hat{P}_{ijkh} & \text{if $i$ odd, $j$  odd, $k$  odd, $h$  odd},\\
 0 & \text{otherwise},
\end{cases}
\end{aligned}\\
\begin{aligned}
&\int_0^{2\pi}\cos^{i+k+1}(\theta)\sin^{j+h+1}(\theta)d\theta\\
&=\begin{cases}
\pi Q_{ijkh} & \text{if $i$  even, $j$  even, $k$  odd, $h$  odd},\\
& \text{or $i$ odd, $j$  odd, $k$  even, $h$  even},\\
\pi \tilde{Q}_{ijkh} & \text{if $i$  even, $j$  odd, $k$ odd, $h$  even},\\
& \text{or $i$ odd, $j$  even, $k$ even, $h$  odd},\\
 0 & \text{otherwise},
\end{cases}
\end{aligned} \\
\begin{aligned}
&\int_0^{2\pi}\cos^{i+k}(\theta)\sin^{j+h+3}(\theta)d\theta \\
&=\begin{cases}
\pi R_{ijkh} & \text{if $i$  even, $j$  even, $k$ even, $h$  odd},\\
& \text{or $i$  even, $j$  odd, $k$ even, $h$  even},\\
\pi \tilde{R}_{ijkh} & \text{if $i$ odd, $j$  even, $k$ odd, $h$  odd},\\
& \text{or $i$  odd, $j$  odd, $k$ odd, $h$  even},\\
 0 & \text{otherwise},
\end{cases}
\end{aligned}\\
\begin{aligned}
&\int_0^{2\pi}\cos^{i+k+2}(\theta)\sin^{j+h+2}(\theta)d\theta \\
&=\begin{cases}
\pi T_{ijkh} & \text{if $i$  even, $j$  odd, $k$  even, $h$  odd},\\
\pi \tilde{T}_{ijkh} & \text{if $i$ odd, $j$  even, $k$  is odd, $h$ is even},\\
\pi \hat{T}_{ijkh} & \text{if $i$ odd, $j$  odd, $k$  odd, $h$  odd},\\
 0 & \text{otherwise},
\end{cases}
\end{aligned}\\
\begin{aligned}
&\int_0^{2\pi}\cos^{i+k+2}(\theta)\sin^{j+h+1}(\theta)d\theta \\
&=\begin{cases}
\pi U_{ijkh} & \text{if $i$  even, $j$  even, $k$ even, $h$ odd},\\
& \text{or $i$ even, $j$  odd, $k$  even, $h$  even},\\
\pi \tilde{U}_{ijkh} & \text{if $i$ odd, $j$  even, $k$  odd, $h$  odd},\\
& \text{or $i$  odd, $j$  odd, $k$ odd, $h$  even},\\
 0 & \text{otherwise},
\end{cases}
\end{aligned}\\
\begin{aligned}
&\int_0^{2\pi}\cos^{i+k+1}(\theta)\sin^{j+h+3}(\theta)d\theta \\
&= \begin{cases}
\pi V_{ijkh} & \text{if $i$  even, $j$  even, $k$  odd, $h$  odd},\\
\pi \tilde{V}_{ijkh} & \text{if $i$  even, $j$  odd, $k$ odd, $h$  even},\\
& \text{or $i$ odd, $j$  even, $k$ even, $h$  odd},\\
0 & \text{otherwise},
\end{cases}
\end{aligned}
\end{gather*}
where
$M_{ijkh}$, $\tilde{M}_{ijkh}$, $N_{ijkh}$, $\tilde{N}_{ijkh}$,
$\hat{N}_{ijkh}$, $Q_{ijkh}$, $\tilde{Q}_{ijkh}$, $P_{ijkh}$, $\tilde{P}_{ijkh}$,
$\hat{P}_{ijkh}$, $R_{ijkh}$, $\tilde{R}{ijkh}$, $T_{ijkh}$,
$\tilde{T}_{ijkh}$, $\hat{T}_{ijkh}$, $ U_{ijkh}$, $\tilde{U}_{ijkh}$,
$V_{ijkh}$, $\tilde{V}_{ijkh}$
are non-zero constants.

\subsection*{Conclusion}

This article concerns the second part of the 16th Hilbert problem in which
we study the bifurcation of limit cycles from the periodic orbits of a
linear center when we perturb it inside a general class of all polynomial
differential systems. We provide an accurate upper bound of the maximum
number of limit cycles that this class of systems can have and we give an
example which illustrates that this bound can be reached.
We would like to stress that although this work ultimately focuses
a general class of polynomial differential systems, the
``Averaging method'' summarized in section 2 can be adapted to the study of
 other polynomial systems. However, the difficulty lies in the complicated
form of the averaged function of second order obtained when the first order
one is identically zero.
The bifurcation of limit cycles from isochronous center will be the
subject of a future work.

\subsection*{Acknowledgements}
We want to thank the reviewers all their comments and suggestions which
help us to improve the presentation of this article.


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\end{document}