所求方程： $$ \ddot{x} + \omega_0^2 x + Ax^2 +B x^3 + ... = 0 $$ 假设： $$ x(t)=\sum_{n=-\infty}^{\infty} f_n e^{i n \omega t}\\ \omega = \omega_0 + \omega_1 +\omega_2 +\cdots \\ f_n=f_n^{(0)}+f_n^{(1)}+f_n^{(2)}+\cdots $$ 用阶数命名AB等参数： $$ \omega_0^2\to A_0\\ A\to A_1 \\ B\to A_2 \\ C\to A_3 $$

$$ \omega_0^2 x <<Ax^2 <<Bx^3 $$

import numpy,scipy,sympy
from IPython.display import display, Markdown,Latex,display_latex
# from utils import *
import sympy
import itertools,functools
from sympy.abc import a,b,x,p,n
MAXLEN=10
iteration=8
def getidx(i):
    return i+MAXLEN
#-------------符号定义----------------------
omegas=sympy.symbols(','.join(["omega_{}".format(i) for i in range(iteration)]),positive=True)
sym_f =[[sympy.symbols("f_{{{}}}^{{({})}}".format(i,j)) for i in range(-MAXLEN,MAXLEN+1)] for j in range(iteration)] #所有符号
f_sym=[sympy.symbols("f^{{({})}}".format(i)) for i in range(iteration)]
params=[omegas[0]**2]+[sympy.symbols("A_{{{}}}".format(i)) for i in range(1,iteration)] #Ax^2,Bx^3,...
res_f=[[0 for i in range(-MAXLEN,MAXLEN+1)] for j in range(iteration)] #存储结果
res_f[0][getidx(1)] = a       #初值
res_f[0][getidx(-1)] = a.conjugate()
res_conv_f =[ [[0 for i in range(-MAXLEN,MAXLEN+1)] for j in range(iteration)] for k in range(iteration)]
res_conv_f[0][0]=res_f[0][:]
# res_conv_f[i][k][getidx(n)] #i阶项,k次卷积，第n个值
#p 为阶数，计算时为了求出统一阶数，方便合并同阶项，用p为宗量构造多项式，
# 即 omega=omega_0 +omega_1 + ... ---->    omega_0 p^0 + omega_1 p^1 + ...
f_sym_tmp = sympy.Poly(reversed(f_sym),p)  #多项式，构造用
omega_tmp=sympy.Poly(reversed(omegas),p) #omega对应多项式

# 构建非线性项：
nonlin=0
for i in range(iteration):
    nonlin+=p**i * f_sym_tmp**(i+1) *params[i]
coeffs_tmp = nonlin-n**2 * f_sym_tmp*omega_tmp**2  #e^{in\omega t}的系数的表达式，对应n的每一项都应该为零
coeffs=[]
for i in range(iteration):
    coeffs.append(coeffs_tmp.coeff_monomial(p**i)) #coeffs[i]: p**i为 order i 的系数表达式，最高取到多少阶是准确的？iteration order！
# coeffs[3]

#-------------end-------------------------




#---------------------函数定义-----------------------------

def dconv(a,b,ret=None): #discrete convolution
    if ret==[]: 
        append,ref=True,True
    elif ret==None: #return the coeffsult. Do not save in ret
        append,ref=False,False
    else:
        append,ref=False,True
    tmp=[]
    for u in range(-MAXLEN,MAXLEN+1):
        s=0
        for n in range(max(u-MAXLEN,-MAXLEN),min(u+MAXLEN,MAXLEN)+1):
            s+= a[getidx(n)]*b[getidx(u-n)] 
        if ref:
            if append:
                ret.append(s)
            else:
                ret[getidx(u)]=s
        else:
            tmp.append(s)
    return tmp
def dconv_n(a,b,n): #计算离散卷积的第n项
    s=0
    assert n>=-MAXLEN and n<= MAXLEN
    for i in range(max(n-MAXLEN,-MAXLEN),min(n+MAXLEN,MAXLEN)+1):
        s+= a[getidx(i)]*b[getidx(n-i)]
    return s
def kPower_dconv_n(a,k,n):
    assert n>= -MAXLEN and n<= MAXLEN 
    cp=a[:]
    for i in range(k-1):
        cp=dconv(cp,a)
    return cp[n]
def gen_dconv(a):
    cp=a[:]
    while True:
        yield cp
        cp=dconv(cp,a)

def show(f,upperidx=None): #显示输出
    if not  upperidx: #不输出上标
        for i in range(MAXLEN+1):
            if f[getidx(i)]==0:
                continue
            if i==0:
                ltx=sympy.latex(f[getidx(0)])
                display(Latex(f"$f_0={ltx}$"))
            else:
                ltx=sympy.latex(f[getidx(i)])
                ltx2=sympy.latex(f[getidx(-i)])
                display(Latex(f"$f_{{{i}}}={ltx},\qquad f_{{{-i}}}={ltx2}$"))
    else:
        display(Markdown(f"# 第{upperidx}阶："))
        for i in range(MAXLEN+1):
            if f[getidx(i)]==0:
                continue
            if i==0:
                ltx=sympy.latex(f[getidx(0)])
                display(Latex(f"$f_0^{{{upperidx}}}={ltx}$"))
            else:
                ltx=sympy.latex(f[getidx(i)])
                ltx2=sympy.latex(f[getidx(-i)])
                display(Latex(f"$f_{{{i}}}^{{{upperidx}}}={ltx},\qquad f_{{{-i}}}^{{{upperidx}}}={ltx2}$"))
        
def evaluate(expr):
    def addlist(*args):
        ret=[]
        
        for i in range(-MAXLEN,MAXLEN+1):
            s=0
            for ele in args:
                s+= ele[getidx(i)]
            ret.append(s)
            # ret.append(sum([ele[getidx(i)] for ele in args]))
        return ret 
    def eval(expr):
        if expr.func == sympy.Add: #应该为表达式树的最顶层，所有列表对应项加起来
            args=[]
            for i in range(len(expr.args)):
                args.append(eval(expr.args[i]))
            return addlist(*args)
        elif expr.func == sympy.Mul: #考虑(f^(k1))^a1 * (f^(k2))^a2 *... 项
            length= len(expr.args)
            is_f = [False for _ in range(length)]
            pre_calc=[None for _ in range(length)]
            for idx,arg in enumerate(expr.args):
                if arg.func == sympy.Symbol and arg in f_sym: #一次项
                    order=f_sym.index(arg)
                    is_f[idx]=True
                    pre_calc[idx]=res_conv_f[order][0][:]
                elif arg.func == sympy.Pow and arg.args[0] in f_sym: #高阶卷积项
                    order=f_sym.index(arg.args[0])
                    power=arg.args[1]
                    is_f[idx]=True
                    pre_calc[idx]=res_conv_f[order][power-1][:]
            to_do_conv_set=[]
            counter_set=[]
            for i in range(length):
                if is_f[i] :
                    to_do_conv_set.append(pre_calc[i])
                else:
                    counter_set.append(expr.args[i])
            ret = functools.reduce(lambda x,y:dconv(x,y),to_do_conv_set) #列表
            const=functools.reduce(lambda x,y:x*y,counter_set,1)
            return [const.subs(n,i) * ret[getidx(i)] for i in range(-MAXLEN,MAXLEN+1)]
        else:
            raise NotImplementedError("Not implemented.")
    return eval(expr)
def evaluate_omega(expr,symbol):
    k=expr.coeff(symbol).subs(n,1)
    b=(expr-expr.coeff(symbol)*symbol).subs(n,1)

    
    k_=evaluate(k)[getidx(1)] #(-n^2+1) \omega_0^2
    b_=evaluate(b)[getidx(1)]

    display(Markdown(f"$k\omega+b=0$，其中："))
    display(Latex(f"$$k={sympy.latex(k)}\\\\b={sympy.latex(b)}$$"))
    display(Latex(f"after evaluation:$$k={sympy.latex(k_)}\\\\b={sympy.latex(b_)}$$"))
    return sympy.simplify(-b_/k_)
def evaluate_f(expr,symbol):

    b=expr-expr.coeff(symbol)*symbol
    k_ = [(-i**2+1)*params[0] for i in range(-MAXLEN,MAXLEN+1)] #系数是固定的，每一阶都相同
    b_:list=evaluate(b)
    fn=[]
    for i in range(-MAXLEN,MAXLEN+1):
        idx=getidx(i)
        if abs(i)==1:
            fn.append(0)
        else:
            fn.append(sympy.expand(-b_[idx]/k_[idx]))
    return fn 
def evaluate_order(order):
    #计算前一项的所有卷积：
    display(Markdown(f"# 第{order}阶计算："))
    pre=order-1
    generator = gen_dconv(res_f[pre][:])
    for i in range(iteration):
        res_conv_f[pre][i] =next(generator)
    #计算omegas[order]的值 n=1
    display(Markdown("### $\omega$的计算："))
    omega=evaluate_omega(coeffs[order],omegas[order])
    display(Markdown(f"### $\omega_{order}={sympy.latex(omega)}$"))
    for i in range(iteration): #对于全部系数：
        coeffs[i] =coeffs[i].subs(omegas[order],omega) #代入omega值
    display(Markdown(f"### $f^{{({order})}}$的计算："))
    res_f[order]=evaluate_f(coeffs[order],f_sym[order])
    show(res_f[order])
    
    
#f0 的卷积除以f0不是降幂次！所以不能用solve直接除

#----------------end----------------------

# main:
for i in range(1,iteration):
    evaluate_order(i)

第1阶计算：

$\omega$的计算：

$k\omega+b=0$，其中：

$$k=- 2 f^{(0)} \omega_{0}\\b=A_{1} \left(f^{(0)}\right)^{2}$$

after evaluation:$$k=- 2 a \omega_{0}\\b=0$$

$\omega_1=0$

$f^{(1)}$的计算：

$f_0=- \frac{2 A_{1} a \overline{a}}{\omega_{0}^{2}}$

$f_{2}=\frac{A_{1} a^{2}}{3 \omega_{0}^{2}},\qquad f_{-2}=\frac{A_{1} \overline{a}^{2}}{3 \omega_{0}^{2}}$

第2阶计算：

$\omega$的计算：

$k\omega+b=0$，其中：

$$k=- 2 f^{(0)} \omega_{0}\\b=2 A_{1} f^{(0)} f^{(1)} + A_{2} \left(f^{(0)}\right)^{3}$$

after evaluation:$$k=- 2 a \omega_{0}\\b=- \frac{10 A_{1}^{2} a^{2} \overline{a}}{3 \omega_{0}^{2}} + 3 A_{2} a^{2} \overline{a}$$

$\omega_2=\frac{a \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right) \overline{a}}{6 \omega_{0}^{3}}$

$f^{(2)}$的计算：

$f_{3}=\frac{A_{1}^{2} a^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} a^{3}}{8 \omega_{0}^{2}},\qquad f_{-3}=\frac{A_{1}^{2} \overline{a}^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} \overline{a}^{3}}{8 \omega_{0}^{2}}$

第3阶计算：

$\omega$的计算：

$k\omega+b=0$，其中：

$$k=- 2 f^{(0)} \omega_{0}\\b=2 A_{1} f^{(0)} f^{(2)} + A_{1} \left(f^{(1)}\right)^{2} + 3 A_{2} \left(f^{(0)}\right)^{2} f^{(1)} + A_{3} \left(f^{(0)}\right)^{4} - \frac{a f^{(1)} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right) \overline{a}}{3 \omega_{0}^{2}}$$

after evaluation:$$k=- 2 a \omega_{0}\\b=0$$

$\omega_3=0$

$f^{(3)}$的计算：

$f_0=- \frac{38 A_{1}^{3} a^{2} \overline{a}^{2}}{9 \omega_{0}^{6}} + \frac{10 A_{1} A_{2} a^{2} \overline{a}^{2}}{\omega_{0}^{4}} - \frac{6 A_{3} a^{2} \overline{a}^{2}}{\omega_{0}^{2}}$

$f_{2}=\frac{59 A_{1}^{3} a^{3} \overline{a}}{54 \omega_{0}^{6}} - \frac{31 A_{1} A_{2} a^{3} \overline{a}}{12 \omega_{0}^{4}} + \frac{4 A_{3} a^{3} \overline{a}}{3 \omega_{0}^{2}},\qquad f_{-2}=\frac{59 A_{1}^{3} a \overline{a}^{3}}{54 \omega_{0}^{6}} - \frac{31 A_{1} A_{2} a \overline{a}^{3}}{12 \omega_{0}^{4}} + \frac{4 A_{3} a \overline{a}^{3}}{3 \omega_{0}^{2}}$

$f_{4}=\frac{A_{1}^{3} a^{4}}{54 \omega_{0}^{6}} + \frac{A_{1} A_{2} a^{4}}{12 \omega_{0}^{4}} + \frac{A_{3} a^{4}}{15 \omega_{0}^{2}},\qquad f_{-4}=\frac{A_{1}^{3} \overline{a}^{4}}{54 \omega_{0}^{6}} + \frac{A_{1} A_{2} \overline{a}^{4}}{12 \omega_{0}^{4}} + \frac{A_{3} \overline{a}^{4}}{15 \omega_{0}^{2}}$

第4阶计算：

$\omega$的计算：

$k\omega+b=0$，其中：

$$k=- 2 f^{(0)} \omega_{0}\\b=2 A_{1} f^{(0)} f^{(3)} + 2 A_{1} f^{(1)} f^{(2)} + 3 A_{2} \left(f^{(0)}\right)^{2} f^{(2)} + 3 A_{2} f^{(0)} \left(f^{(1)}\right)^{2} + 4 A_{3} \left(f^{(0)}\right)^{3} f^{(1)} + A_{4} \left(f^{(0)}\right)^{5} - \frac{a^{2} f^{(0)} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right)^{2} \overline{a}^{2}}{36 \omega_{0}^{6}} - \frac{a f^{(2)} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right) \overline{a}}{3 \omega_{0}^{2}}$$

after evaluation:$$k=- 2 a \omega_{0}\\b=\frac{26 A_{1}^{2} A_{2} a^{3} \overline{a}^{2}}{3 \omega_{0}^{4}} + \frac{2 A_{1}^{2} \left(\frac{A_{1}^{2} a^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} a^{3}}{8 \omega_{0}^{2}}\right) \overline{a}^{2}}{3 \omega_{0}^{2}} - \frac{56 A_{1} A_{3} a^{3} \overline{a}^{2}}{3 \omega_{0}^{2}} + 2 A_{1} \left(a \left(- \frac{38 A_{1}^{3} a^{2} \overline{a}^{2}}{9 \omega_{0}^{6}} + \frac{10 A_{1} A_{2} a^{2} \overline{a}^{2}}{\omega_{0}^{4}} - \frac{6 A_{3} a^{2} \overline{a}^{2}}{\omega_{0}^{2}}\right) + \left(\frac{59 A_{1}^{3} a^{3} \overline{a}}{54 \omega_{0}^{6}} - \frac{31 A_{1} A_{2} a^{3} \overline{a}}{12 \omega_{0}^{4}} + \frac{4 A_{3} a^{3} \overline{a}}{3 \omega_{0}^{2}}\right) \overline{a}\right) + 3 A_{2} \left(\frac{A_{1}^{2} a^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} a^{3}}{8 \omega_{0}^{2}}\right) \overline{a}^{2} + 10 A_{4} a^{3} \overline{a}^{2} - \frac{a^{3} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right)^{2} \overline{a}^{2}}{36 \omega_{0}^{6}}$$

$\omega_4=\frac{a^{2} \left(- 1940 A_{1}^{4} + 6228 A_{1}^{2} A_{2} \omega_{0}^{2} - 6048 A_{1} A_{3} \omega_{0}^{4} - 405 A_{2}^{2} \omega_{0}^{4} + 2160 A_{4} \omega_{0}^{6}\right) \overline{a}^{2}}{432 \omega_{0}^{7}}$

$f^{(4)}$的计算：

$f_{3}=\frac{79 A_{1}^{4} a^{4} \overline{a}}{144 \omega_{0}^{8}} - \frac{43 A_{1}^{2} A_{2} a^{4} \overline{a}}{48 \omega_{0}^{6}} - \frac{3 A_{1} A_{3} a^{4} \overline{a}}{20 \omega_{0}^{4}} - \frac{21 A_{2}^{2} a^{4} \overline{a}}{64 \omega_{0}^{4}} + \frac{5 A_{4} a^{4} \overline{a}}{8 \omega_{0}^{2}},\qquad f_{-3}=\frac{79 A_{1}^{4} a \overline{a}^{4}}{144 \omega_{0}^{8}} - \frac{43 A_{1}^{2} A_{2} a \overline{a}^{4}}{48 \omega_{0}^{6}} - \frac{3 A_{1} A_{3} a \overline{a}^{4}}{20 \omega_{0}^{4}} - \frac{21 A_{2}^{2} a \overline{a}^{4}}{64 \omega_{0}^{4}} + \frac{5 A_{4} a \overline{a}^{4}}{8 \omega_{0}^{2}}$

$f_{5}=\frac{5 A_{1}^{4} a^{5}}{1296 \omega_{0}^{8}} + \frac{5 A_{1}^{2} A_{2} a^{5}}{144 \omega_{0}^{6}} + \frac{11 A_{1} A_{3} a^{5}}{180 \omega_{0}^{4}} + \frac{A_{2}^{2} a^{5}}{64 \omega_{0}^{4}} + \frac{A_{4} a^{5}}{24 \omega_{0}^{2}},\qquad f_{-5}=\frac{5 A_{1}^{4} \overline{a}^{5}}{1296 \omega_{0}^{8}} + \frac{5 A_{1}^{2} A_{2} \overline{a}^{5}}{144 \omega_{0}^{6}} + \frac{11 A_{1} A_{3} \overline{a}^{5}}{180 \omega_{0}^{4}} + \frac{A_{2}^{2} \overline{a}^{5}}{64 \omega_{0}^{4}} + \frac{A_{4} \overline{a}^{5}}{24 \omega_{0}^{2}}$

第5阶计算：

$\omega$的计算：

$k\omega+b=0$，其中：

$$k=- 2 f^{(0)} \omega_{0}\\b=2 A_{1} f^{(0)} f^{(4)} + 2 A_{1} f^{(1)} f^{(3)} + A_{1} \left(f^{(2)}\right)^{2} + 3 A_{2} \left(f^{(0)}\right)^{2} f^{(3)} + 6 A_{2} f^{(0)} f^{(1)} f^{(2)} + A_{2} \left(f^{(1)}\right)^{3} + 4 A_{3} \left(f^{(0)}\right)^{3} f^{(2)} + 6 A_{3} \left(f^{(0)}\right)^{2} \left(f^{(1)}\right)^{2} + 5 A_{4} \left(f^{(0)}\right)^{4} f^{(1)} + A_{5} \left(f^{(0)}\right)^{6} - \frac{a^{2} f^{(1)} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right)^{2} \overline{a}^{2}}{36 \omega_{0}^{6}} - \frac{a^{2} f^{(1)} \left(- 1940 A_{1}^{4} + 6228 A_{1}^{2} A_{2} \omega_{0}^{2} - 6048 A_{1} A_{3} \omega_{0}^{4} - 405 A_{2}^{2} \omega_{0}^{4} + 2160 A_{4} \omega_{0}^{6}\right) \overline{a}^{2}}{216 \omega_{0}^{6}} - \frac{a f^{(3)} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right) \overline{a}}{3 \omega_{0}^{2}}$$

after evaluation:$$k=- 2 a \omega_{0}\\b=0$$

$\omega_5=0$

$f^{(5)}$的计算：

$f_0=- \frac{11897 A_{1}^{5} a^{3} \overline{a}^{3}}{648 \omega_{0}^{10}} + \frac{5125 A_{1}^{3} A_{2} a^{3} \overline{a}^{3}}{72 \omega_{0}^{8}} - \frac{550 A_{1}^{2} A_{3} a^{3} \overline{a}^{3}}{9 \omega_{0}^{6}} - \frac{1441 A_{1} A_{2}^{2} a^{3} \overline{a}^{3}}{32 \omega_{0}^{6}} + \frac{140 A_{1} A_{4} a^{3} \overline{a}^{3}}{3 \omega_{0}^{4}} + \frac{27 A_{2} A_{3} a^{3} \overline{a}^{3}}{\omega_{0}^{4}} - \frac{20 A_{5} a^{3} \overline{a}^{3}}{\omega_{0}^{2}}$

$f_{2}=\frac{1207 A_{1}^{5} a^{4} \overline{a}^{2}}{216 \omega_{0}^{10}} - \frac{4813 A_{1}^{3} A_{2} a^{4} \overline{a}^{2}}{216 \omega_{0}^{8}} + \frac{5087 A_{1}^{2} A_{3} a^{4} \overline{a}^{2}}{270 \omega_{0}^{6}} + \frac{1387 A_{1} A_{2}^{2} a^{4} \overline{a}^{2}}{96 \omega_{0}^{6}} - \frac{485 A_{1} A_{4} a^{4} \overline{a}^{2}}{36 \omega_{0}^{4}} - \frac{81 A_{2} A_{3} a^{4} \overline{a}^{2}}{10 \omega_{0}^{4}} + \frac{5 A_{5} a^{4} \overline{a}^{2}}{\omega_{0}^{2}},\qquad f_{-2}=\frac{1207 A_{1}^{5} a^{2} \overline{a}^{4}}{216 \omega_{0}^{10}} - \frac{4813 A_{1}^{3} A_{2} a^{2} \overline{a}^{4}}{216 \omega_{0}^{8}} + \frac{5087 A_{1}^{2} A_{3} a^{2} \overline{a}^{4}}{270 \omega_{0}^{6}} + \frac{1387 A_{1} A_{2}^{2} a^{2} \overline{a}^{4}}{96 \omega_{0}^{6}} - \frac{485 A_{1} A_{4} a^{2} \overline{a}^{4}}{36 \omega_{0}^{4}} - \frac{81 A_{2} A_{3} a^{2} \overline{a}^{4}}{10 \omega_{0}^{4}} + \frac{5 A_{5} a^{2} \overline{a}^{4}}{\omega_{0}^{2}}$

$f_{4}=\frac{89 A_{1}^{5} a^{5} \overline{a}}{486 \omega_{0}^{10}} + \frac{A_{1}^{3} A_{2} a^{5} \overline{a}}{9 \omega_{0}^{8}} - \frac{A_{1}^{2} A_{3} a^{5} \overline{a}}{9 \omega_{0}^{6}} - \frac{7 A_{1} A_{2}^{2} a^{5} \overline{a}}{8 \omega_{0}^{6}} - \frac{2 A_{1} A_{4} a^{5} \overline{a}}{15 \omega_{0}^{4}} + \frac{9 A_{2} A_{3} a^{5} \overline{a}}{50 \omega_{0}^{4}} + \frac{2 A_{5} a^{5} \overline{a}}{5 \omega_{0}^{2}},\qquad f_{-4}=\frac{89 A_{1}^{5} a \overline{a}^{5}}{486 \omega_{0}^{10}} + \frac{A_{1}^{3} A_{2} a \overline{a}^{5}}{9 \omega_{0}^{8}} - \frac{A_{1}^{2} A_{3} a \overline{a}^{5}}{9 \omega_{0}^{6}} - \frac{7 A_{1} A_{2}^{2} a \overline{a}^{5}}{8 \omega_{0}^{6}} - \frac{2 A_{1} A_{4} a \overline{a}^{5}}{15 \omega_{0}^{4}} + \frac{9 A_{2} A_{3} a \overline{a}^{5}}{50 \omega_{0}^{4}} + \frac{2 A_{5} a \overline{a}^{5}}{5 \omega_{0}^{2}}$

$f_{6}=\frac{A_{1}^{5} a^{6}}{1296 \omega_{0}^{10}} + \frac{5 A_{1}^{3} A_{2} a^{6}}{432 \omega_{0}^{8}} + \frac{A_{1}^{2} A_{3} a^{6}}{30 \omega_{0}^{6}} + \frac{A_{1} A_{2}^{2} a^{6}}{64 \omega_{0}^{6}} + \frac{A_{1} A_{4} a^{6}}{20 \omega_{0}^{4}} + \frac{A_{2} A_{3} a^{6}}{50 \omega_{0}^{4}} + \frac{A_{5} a^{6}}{35 \omega_{0}^{2}},\qquad f_{-6}=\frac{A_{1}^{5} \overline{a}^{6}}{1296 \omega_{0}^{10}} + \frac{5 A_{1}^{3} A_{2} \overline{a}^{6}}{432 \omega_{0}^{8}} + \frac{A_{1}^{2} A_{3} \overline{a}^{6}}{30 \omega_{0}^{6}} + \frac{A_{1} A_{2}^{2} \overline{a}^{6}}{64 \omega_{0}^{6}} + \frac{A_{1} A_{4} \overline{a}^{6}}{20 \omega_{0}^{4}} + \frac{A_{2} A_{3} \overline{a}^{6}}{50 \omega_{0}^{4}} + \frac{A_{5} \overline{a}^{6}}{35 \omega_{0}^{2}}$

第6阶计算：

$\omega$的计算：

$k\omega+b=0$，其中：

$$k=- 2 f^{(0)} \omega_{0}\\b=2 A_{1} f^{(0)} f^{(5)} + 2 A_{1} f^{(1)} f^{(4)} + 2 A_{1} f^{(2)} f^{(3)} + 3 A_{2} \left(f^{(0)}\right)^{2} f^{(4)} + 6 A_{2} f^{(0)} f^{(1)} f^{(3)} + 3 A_{2} f^{(0)} \left(f^{(2)}\right)^{2} + 3 A_{2} \left(f^{(1)}\right)^{2} f^{(2)} + 4 A_{3} \left(f^{(0)}\right)^{3} f^{(3)} + 12 A_{3} \left(f^{(0)}\right)^{2} f^{(1)} f^{(2)} + 4 A_{3} f^{(0)} \left(f^{(1)}\right)^{3} + 5 A_{4} \left(f^{(0)}\right)^{4} f^{(2)} + 10 A_{4} \left(f^{(0)}\right)^{3} \left(f^{(1)}\right)^{2} + 6 A_{5} \left(f^{(0)}\right)^{5} f^{(1)} + A_{6} \left(f^{(0)}\right)^{7} - \frac{a^{3} f^{(0)} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right) \left(- 1940 A_{1}^{4} + 6228 A_{1}^{2} A_{2} \omega_{0}^{2} - 6048 A_{1} A_{3} \omega_{0}^{4} - 405 A_{2}^{2} \omega_{0}^{4} + 2160 A_{4} \omega_{0}^{6}\right) \overline{a}^{3}}{1296 \omega_{0}^{10}} - \frac{a^{2} f^{(2)} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right)^{2} \overline{a}^{2}}{36 \omega_{0}^{6}} - \frac{a^{2} f^{(2)} \left(- 1940 A_{1}^{4} + 6228 A_{1}^{2} A_{2} \omega_{0}^{2} - 6048 A_{1} A_{3} \omega_{0}^{4} - 405 A_{2}^{2} \omega_{0}^{4} + 2160 A_{4} \omega_{0}^{6}\right) \overline{a}^{2}}{216 \omega_{0}^{6}} - \frac{a f^{(4)} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right) \overline{a}}{3 \omega_{0}^{2}}$$

after evaluation:$$k=- 2 a \omega_{0}\\b=- \frac{188 A_{1}^{3} A_{3} a^{4} \overline{a}^{3}}{9 \omega_{0}^{6}} + \frac{670 A_{1}^{2} A_{4} a^{4} \overline{a}^{3}}{9 \omega_{0}^{4}} + \frac{2 A_{1}^{2} \left(\frac{79 A_{1}^{4} a^{4} \overline{a}}{144 \omega_{0}^{8}} - \frac{43 A_{1}^{2} A_{2} a^{4} \overline{a}}{48 \omega_{0}^{6}} - \frac{3 A_{1} A_{3} a^{4} \overline{a}}{20 \omega_{0}^{4}} - \frac{21 A_{2}^{2} a^{4} \overline{a}}{64 \omega_{0}^{4}} + \frac{5 A_{4} a^{4} \overline{a}}{8 \omega_{0}^{2}}\right) \overline{a}^{2}}{3 \omega_{0}^{2}} - \frac{90 A_{1} A_{5} a^{4} \overline{a}^{3}}{\omega_{0}^{2}} + 2 A_{1} \left(a \left(- \frac{11897 A_{1}^{5} a^{3} \overline{a}^{3}}{648 \omega_{0}^{10}} + \frac{5125 A_{1}^{3} A_{2} a^{3} \overline{a}^{3}}{72 \omega_{0}^{8}} - \frac{550 A_{1}^{2} A_{3} a^{3} \overline{a}^{3}}{9 \omega_{0}^{6}} - \frac{1441 A_{1} A_{2}^{2} a^{3} \overline{a}^{3}}{32 \omega_{0}^{6}} + \frac{140 A_{1} A_{4} a^{3} \overline{a}^{3}}{3 \omega_{0}^{4}} + \frac{27 A_{2} A_{3} a^{3} \overline{a}^{3}}{\omega_{0}^{4}} - \frac{20 A_{5} a^{3} \overline{a}^{3}}{\omega_{0}^{2}}\right) + \left(\frac{1207 A_{1}^{5} a^{4} \overline{a}^{2}}{216 \omega_{0}^{10}} - \frac{4813 A_{1}^{3} A_{2} a^{4} \overline{a}^{2}}{216 \omega_{0}^{8}} + \frac{5087 A_{1}^{2} A_{3} a^{4} \overline{a}^{2}}{270 \omega_{0}^{6}} + \frac{1387 A_{1} A_{2}^{2} a^{4} \overline{a}^{2}}{96 \omega_{0}^{6}} - \frac{485 A_{1} A_{4} a^{4} \overline{a}^{2}}{36 \omega_{0}^{4}} - \frac{81 A_{2} A_{3} a^{4} \overline{a}^{2}}{10 \omega_{0}^{4}} + \frac{5 A_{5} a^{4} \overline{a}^{2}}{\omega_{0}^{2}}\right) \overline{a}\right) + 2 A_{1} \left(\left(\frac{A_{1}^{2} a^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} a^{3}}{8 \omega_{0}^{2}}\right) \left(\frac{59 A_{1}^{3} a \overline{a}^{3}}{54 \omega_{0}^{6}} - \frac{31 A_{1} A_{2} a \overline{a}^{3}}{12 \omega_{0}^{4}} + \frac{4 A_{3} a \overline{a}^{3}}{3 \omega_{0}^{2}}\right) + \left(\frac{A_{1}^{2} \overline{a}^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} \overline{a}^{3}}{8 \omega_{0}^{2}}\right) \left(\frac{A_{1}^{3} a^{4}}{54 \omega_{0}^{6}} + \frac{A_{1} A_{2} a^{4}}{12 \omega_{0}^{4}} + \frac{A_{3} a^{4}}{15 \omega_{0}^{2}}\right)\right) + 6 A_{2} a \left(\frac{A_{1}^{2} a^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} a^{3}}{8 \omega_{0}^{2}}\right) \left(\frac{A_{1}^{2} \overline{a}^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} \overline{a}^{3}}{8 \omega_{0}^{2}}\right) + 3 A_{2} \left(\frac{A_{1}^{2} a^{4} \left(\frac{A_{1}^{2} \overline{a}^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} \overline{a}^{3}}{8 \omega_{0}^{2}}\right)}{9 \omega_{0}^{4}} - \frac{4 A_{1}^{2} a \left(\frac{A_{1}^{2} a^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} a^{3}}{8 \omega_{0}^{2}}\right) \overline{a}^{3}}{3 \omega_{0}^{4}}\right) + 6 A_{2} \left(\frac{A_{1} a^{3} \left(\frac{59 A_{1}^{3} a \overline{a}^{3}}{54 \omega_{0}^{6}} - \frac{31 A_{1} A_{2} a \overline{a}^{3}}{12 \omega_{0}^{4}} + \frac{4 A_{3} a \overline{a}^{3}}{3 \omega_{0}^{2}}\right)}{3 \omega_{0}^{2}} - \frac{5 A_{1} a^{2} \left(- \frac{38 A_{1}^{3} a^{2} \overline{a}^{2}}{9 \omega_{0}^{6}} + \frac{10 A_{1} A_{2} a^{2} \overline{a}^{2}}{\omega_{0}^{4}} - \frac{6 A_{3} a^{2} \overline{a}^{2}}{\omega_{0}^{2}}\right) \overline{a}}{3 \omega_{0}^{2}} - \frac{5 A_{1} a \left(\frac{59 A_{1}^{3} a^{3} \overline{a}}{54 \omega_{0}^{6}} - \frac{31 A_{1} A_{2} a^{3} \overline{a}}{12 \omega_{0}^{4}} + \frac{4 A_{3} a^{3} \overline{a}}{3 \omega_{0}^{2}}\right) \overline{a}^{2}}{3 \omega_{0}^{2}} + \frac{A_{1} \left(\frac{A_{1}^{3} a^{4}}{54 \omega_{0}^{6}} + \frac{A_{1} A_{2} a^{4}}{12 \omega_{0}^{4}} + \frac{A_{3} a^{4}}{15 \omega_{0}^{2}}\right) \overline{a}^{3}}{3 \omega_{0}^{2}}\right) + 3 A_{2} \left(\frac{79 A_{1}^{4} a^{4} \overline{a}}{144 \omega_{0}^{8}} - \frac{43 A_{1}^{2} A_{2} a^{4} \overline{a}}{48 \omega_{0}^{6}} - \frac{3 A_{1} A_{3} a^{4} \overline{a}}{20 \omega_{0}^{4}} - \frac{21 A_{2}^{2} a^{4} \overline{a}}{64 \omega_{0}^{4}} + \frac{5 A_{4} a^{4} \overline{a}}{8 \omega_{0}^{2}}\right) \overline{a}^{2} + 12 A_{3} \left(\frac{A_{1} a^{4} \left(\frac{A_{1}^{2} \overline{a}^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} \overline{a}^{3}}{8 \omega_{0}^{2}}\right)}{3 \omega_{0}^{2}} - \frac{4 A_{1} a \left(\frac{A_{1}^{2} a^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} a^{3}}{8 \omega_{0}^{2}}\right) \overline{a}^{3}}{3 \omega_{0}^{2}}\right) + 4 A_{3} \left(a^{3} \left(\frac{59 A_{1}^{3} a \overline{a}^{3}}{54 \omega_{0}^{6}} - \frac{31 A_{1} A_{2} a \overline{a}^{3}}{12 \omega_{0}^{4}} + \frac{4 A_{3} a \overline{a}^{3}}{3 \omega_{0}^{2}}\right) + 3 a^{2} \left(- \frac{38 A_{1}^{3} a^{2} \overline{a}^{2}}{9 \omega_{0}^{6}} + \frac{10 A_{1} A_{2} a^{2} \overline{a}^{2}}{\omega_{0}^{4}} - \frac{6 A_{3} a^{2} \overline{a}^{2}}{\omega_{0}^{2}}\right) \overline{a} + 3 a \left(\frac{59 A_{1}^{3} a^{3} \overline{a}}{54 \omega_{0}^{6}} - \frac{31 A_{1} A_{2} a^{3} \overline{a}}{12 \omega_{0}^{4}} + \frac{4 A_{3} a^{3} \overline{a}}{3 \omega_{0}^{2}}\right) \overline{a}^{2} + \left(\frac{A_{1}^{3} a^{4}}{54 \omega_{0}^{6}} + \frac{A_{1} A_{2} a^{4}}{12 \omega_{0}^{4}} + \frac{A_{3} a^{4}}{15 \omega_{0}^{2}}\right) \overline{a}^{3}\right) + 5 A_{4} \left(a^{4} \left(\frac{A_{1}^{2} \overline{a}^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} \overline{a}^{3}}{8 \omega_{0}^{2}}\right) + 4 a \left(\frac{A_{1}^{2} a^{3}}{12 \omega_{0}^{4}} + \frac{A_{2} a^{3}}{8 \omega_{0}^{2}}\right) \overline{a}^{3}\right) + 35 A_{6} a^{4} \overline{a}^{3} - \frac{a^{4} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right) \left(- 1940 A_{1}^{4} + 6228 A_{1}^{2} A_{2} \omega_{0}^{2} - 6048 A_{1} A_{3} \omega_{0}^{4} - 405 A_{2}^{2} \omega_{0}^{4} + 2160 A_{4} \omega_{0}^{6}\right) \overline{a}^{3}}{1296 \omega_{0}^{10}}$$

$\omega_6=\frac{a^{3} \left(- 1035800 A_{1}^{6} + 5015220 A_{1}^{4} A_{2} \omega_{0}^{2} - 4823040 A_{1}^{3} A_{3} \omega_{0}^{4} - 4939650 A_{1}^{2} A_{2}^{2} \omega_{0}^{4} + 4147200 A_{1}^{2} A_{4} \omega_{0}^{6} + 5356800 A_{1} A_{2} A_{3} \omega_{0}^{6} - 3110400 A_{1} A_{5} \omega_{0}^{8} + 49815 A_{2}^{3} \omega_{0}^{6} - 259200 A_{2} A_{4} \omega_{0}^{8} - 1306368 A_{3}^{2} \omega_{0}^{8} + 907200 A_{6} \omega_{0}^{10}\right) \overline{a}^{3}}{51840 \omega_{0}^{11}}$

$f^{(6)}$的计算：

$f_{3}=\frac{19283 A_{1}^{6} a^{5} \overline{a}^{2}}{5184 \omega_{0}^{12}} - \frac{46615 A_{1}^{4} A_{2} a^{5} \overline{a}^{2}}{3456 \omega_{0}^{10}} + \frac{1361 A_{1}^{3} A_{3} a^{5} \overline{a}^{2}}{180 \omega_{0}^{8}} + \frac{1905 A_{1}^{2} A_{2}^{2} a^{5} \overline{a}^{2}}{256 \omega_{0}^{8}} - \frac{467 A_{1}^{2} A_{4} a^{5} \overline{a}^{2}}{360 \omega_{0}^{6}} - \frac{263 A_{1} A_{2} A_{3} a^{5} \overline{a}^{2}}{600 \omega_{0}^{6}} - \frac{17 A_{1} A_{5} a^{5} \overline{a}^{2}}{5 \omega_{0}^{4}} + \frac{417 A_{2}^{3} a^{5} \overline{a}^{2}}{512 \omega_{0}^{6}} - \frac{41 A_{2} A_{4} a^{5} \overline{a}^{2}}{16 \omega_{0}^{4}} - \frac{9 A_{3}^{2} a^{5} \overline{a}^{2}}{10 \omega_{0}^{4}} + \frac{21 A_{6} a^{5} \overline{a}^{2}}{8 \omega_{0}^{2}},\qquad f_{-3}=\frac{19283 A_{1}^{6} a^{2} \overline{a}^{5}}{5184 \omega_{0}^{12}} - \frac{46615 A_{1}^{4} A_{2} a^{2} \overline{a}^{5}}{3456 \omega_{0}^{10}} + \frac{1361 A_{1}^{3} A_{3} a^{2} \overline{a}^{5}}{180 \omega_{0}^{8}} + \frac{1905 A_{1}^{2} A_{2}^{2} a^{2} \overline{a}^{5}}{256 \omega_{0}^{8}} - \frac{467 A_{1}^{2} A_{4} a^{2} \overline{a}^{5}}{360 \omega_{0}^{6}} - \frac{263 A_{1} A_{2} A_{3} a^{2} \overline{a}^{5}}{600 \omega_{0}^{6}} - \frac{17 A_{1} A_{5} a^{2} \overline{a}^{5}}{5 \omega_{0}^{4}} + \frac{417 A_{2}^{3} a^{2} \overline{a}^{5}}{512 \omega_{0}^{6}} - \frac{41 A_{2} A_{4} a^{2} \overline{a}^{5}}{16 \omega_{0}^{4}} - \frac{9 A_{3}^{2} a^{2} \overline{a}^{5}}{10 \omega_{0}^{4}} + \frac{21 A_{6} a^{2} \overline{a}^{5}}{8 \omega_{0}^{2}}$

$f_{5}=\frac{2375 A_{1}^{6} a^{6} \overline{a}}{46656 \omega_{0}^{12}} + \frac{85 A_{1}^{4} A_{2} a^{6} \overline{a}}{384 \omega_{0}^{10}} + \frac{25 A_{1}^{3} A_{3} a^{6} \overline{a}}{108 \omega_{0}^{8}} - \frac{1235 A_{1}^{2} A_{2}^{2} a^{6} \overline{a}}{2304 \omega_{0}^{8}} - \frac{43 A_{1}^{2} A_{4} a^{6} \overline{a}}{216 \omega_{0}^{6}} - \frac{199 A_{1} A_{2} A_{3} a^{6} \overline{a}}{360 \omega_{0}^{6}} - \frac{A_{1} A_{5} a^{6} \overline{a}}{21 \omega_{0}^{4}} - \frac{43 A_{2}^{3} a^{6} \overline{a}}{512 \omega_{0}^{6}} + \frac{A_{2} A_{4} a^{6} \overline{a}}{16 \omega_{0}^{4}} + \frac{23 A_{3}^{2} a^{6} \overline{a}}{90 \omega_{0}^{4}} + \frac{7 A_{6} a^{6} \overline{a}}{24 \omega_{0}^{2}},\qquad f_{-5}=\frac{2375 A_{1}^{6} a \overline{a}^{6}}{46656 \omega_{0}^{12}} + \frac{85 A_{1}^{4} A_{2} a \overline{a}^{6}}{384 \omega_{0}^{10}} + \frac{25 A_{1}^{3} A_{3} a \overline{a}^{6}}{108 \omega_{0}^{8}} - \frac{1235 A_{1}^{2} A_{2}^{2} a \overline{a}^{6}}{2304 \omega_{0}^{8}} - \frac{43 A_{1}^{2} A_{4} a \overline{a}^{6}}{216 \omega_{0}^{6}} - \frac{199 A_{1} A_{2} A_{3} a \overline{a}^{6}}{360 \omega_{0}^{6}} - \frac{A_{1} A_{5} a \overline{a}^{6}}{21 \omega_{0}^{4}} - \frac{43 A_{2}^{3} a \overline{a}^{6}}{512 \omega_{0}^{6}} + \frac{A_{2} A_{4} a \overline{a}^{6}}{16 \omega_{0}^{4}} + \frac{23 A_{3}^{2} a \overline{a}^{6}}{90 \omega_{0}^{4}} + \frac{7 A_{6} a \overline{a}^{6}}{24 \omega_{0}^{2}}$

$f_{7}=\frac{7 A_{1}^{6} a^{7}}{46656 \omega_{0}^{12}} + \frac{35 A_{1}^{4} A_{2} a^{7}}{10368 \omega_{0}^{10}} + \frac{91 A_{1}^{3} A_{3} a^{7}}{6480 \omega_{0}^{8}} + \frac{7 A_{1}^{2} A_{2}^{2} a^{7}}{768 \omega_{0}^{8}} + \frac{149 A_{1}^{2} A_{4} a^{7}}{4320 \omega_{0}^{6}} + \frac{181 A_{1} A_{2} A_{3} a^{7}}{7200 \omega_{0}^{6}} + \frac{3 A_{1} A_{5} a^{7}}{70 \omega_{0}^{4}} + \frac{A_{2}^{3} a^{7}}{512 \omega_{0}^{6}} + \frac{A_{2} A_{4} a^{7}}{64 \omega_{0}^{4}} + \frac{A_{3}^{2} a^{7}}{180 \omega_{0}^{4}} + \frac{A_{6} a^{7}}{48 \omega_{0}^{2}},\qquad f_{-7}=\frac{7 A_{1}^{6} \overline{a}^{7}}{46656 \omega_{0}^{12}} + \frac{35 A_{1}^{4} A_{2} \overline{a}^{7}}{10368 \omega_{0}^{10}} + \frac{91 A_{1}^{3} A_{3} \overline{a}^{7}}{6480 \omega_{0}^{8}} + \frac{7 A_{1}^{2} A_{2}^{2} \overline{a}^{7}}{768 \omega_{0}^{8}} + \frac{149 A_{1}^{2} A_{4} \overline{a}^{7}}{4320 \omega_{0}^{6}} + \frac{181 A_{1} A_{2} A_{3} \overline{a}^{7}}{7200 \omega_{0}^{6}} + \frac{3 A_{1} A_{5} \overline{a}^{7}}{70 \omega_{0}^{4}} + \frac{A_{2}^{3} \overline{a}^{7}}{512 \omega_{0}^{6}} + \frac{A_{2} A_{4} \overline{a}^{7}}{64 \omega_{0}^{4}} + \frac{A_{3}^{2} \overline{a}^{7}}{180 \omega_{0}^{4}} + \frac{A_{6} \overline{a}^{7}}{48 \omega_{0}^{2}}$

第7阶计算：

$\omega$的计算：

$k\omega+b=0$，其中：

$$k=- 2 f^{(0)} \omega_{0}\\b=2 A_{1} f^{(0)} f^{(6)} + 2 A_{1} f^{(1)} f^{(5)} + 2 A_{1} f^{(2)} f^{(4)} + A_{1} \left(f^{(3)}\right)^{2} + 3 A_{2} \left(f^{(0)}\right)^{2} f^{(5)} + 6 A_{2} f^{(0)} f^{(1)} f^{(4)} + 6 A_{2} f^{(0)} f^{(2)} f^{(3)} + 3 A_{2} \left(f^{(1)}\right)^{2} f^{(3)} + 3 A_{2} f^{(1)} \left(f^{(2)}\right)^{2} + 4 A_{3} \left(f^{(0)}\right)^{3} f^{(4)} + 12 A_{3} \left(f^{(0)}\right)^{2} f^{(1)} f^{(3)} + 6 A_{3} \left(f^{(0)}\right)^{2} \left(f^{(2)}\right)^{2} + 12 A_{3} f^{(0)} \left(f^{(1)}\right)^{2} f^{(2)} + A_{3} \left(f^{(1)}\right)^{4} + 5 A_{4} \left(f^{(0)}\right)^{4} f^{(3)} + 20 A_{4} \left(f^{(0)}\right)^{3} f^{(1)} f^{(2)} + 10 A_{4} \left(f^{(0)}\right)^{2} \left(f^{(1)}\right)^{3} + 6 A_{5} \left(f^{(0)}\right)^{5} f^{(2)} + 15 A_{5} \left(f^{(0)}\right)^{4} \left(f^{(1)}\right)^{2} + 7 A_{6} \left(f^{(0)}\right)^{6} f^{(1)} + A_{7} \left(f^{(0)}\right)^{8} - \frac{a^{3} f^{(1)} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right) \left(- 1940 A_{1}^{4} + 6228 A_{1}^{2} A_{2} \omega_{0}^{2} - 6048 A_{1} A_{3} \omega_{0}^{4} - 405 A_{2}^{2} \omega_{0}^{4} + 2160 A_{4} \omega_{0}^{6}\right) \overline{a}^{3}}{1296 \omega_{0}^{10}} - \frac{a^{3} f^{(1)} \left(- 1035800 A_{1}^{6} + 5015220 A_{1}^{4} A_{2} \omega_{0}^{2} - 4823040 A_{1}^{3} A_{3} \omega_{0}^{4} - 4939650 A_{1}^{2} A_{2}^{2} \omega_{0}^{4} + 4147200 A_{1}^{2} A_{4} \omega_{0}^{6} + 5356800 A_{1} A_{2} A_{3} \omega_{0}^{6} - 3110400 A_{1} A_{5} \omega_{0}^{8} + 49815 A_{2}^{3} \omega_{0}^{6} - 259200 A_{2} A_{4} \omega_{0}^{8} - 1306368 A_{3}^{2} \omega_{0}^{8} + 907200 A_{6} \omega_{0}^{10}\right) \overline{a}^{3}}{25920 \omega_{0}^{10}} - \frac{a^{2} f^{(3)} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right)^{2} \overline{a}^{2}}{36 \omega_{0}^{6}} - \frac{a^{2} f^{(3)} \left(- 1940 A_{1}^{4} + 6228 A_{1}^{2} A_{2} \omega_{0}^{2} - 6048 A_{1} A_{3} \omega_{0}^{4} - 405 A_{2}^{2} \omega_{0}^{4} + 2160 A_{4} \omega_{0}^{6}\right) \overline{a}^{2}}{216 \omega_{0}^{6}} - \frac{a f^{(5)} \left(- 10 A_{1}^{2} + 9 A_{2} \omega_{0}^{2}\right) \overline{a}}{3 \omega_{0}^{2}}$$

after evaluation:$$k=- 2 a \omega_{0}\\b=0$$

$\omega_7=0$

$f^{(7)}$的计算：

$f_0=- \frac{1181413 A_{1}^{7} a^{4} \overline{a}^{4}}{11664 \omega_{0}^{14}} + \frac{52397 A_{1}^{5} A_{2} a^{4} \overline{a}^{4}}{96 \omega_{0}^{12}} - \frac{783443 A_{1}^{4} A_{3} a^{4} \overline{a}^{4}}{1620 \omega_{0}^{10}} - \frac{431023 A_{1}^{3} A_{2}^{2} a^{4} \overline{a}^{4}}{576 \omega_{0}^{10}} + \frac{85315 A_{1}^{3} A_{4} a^{4} \overline{a}^{4}}{216 \omega_{0}^{8}} + \frac{36763 A_{1}^{2} A_{2} A_{3} a^{4} \overline{a}^{4}}{40 \omega_{0}^{8}} - \frac{315 A_{1}^{2} A_{5} a^{4} \overline{a}^{4}}{\omega_{0}^{6}} + \frac{24181 A_{1} A_{2}^{3} a^{4} \overline{a}^{4}}{128 \omega_{0}^{8}} - \frac{18935 A_{1} A_{2} A_{4} a^{4} \overline{a}^{4}}{48 \omega_{0}^{6}} - \frac{53152 A_{1} A_{3}^{2} a^{4} \overline{a}^{4}}{225 \omega_{0}^{6}} + \frac{210 A_{1} A_{6} a^{4} \overline{a}^{4}}{\omega_{0}^{4}} - \frac{453 A_{2}^{2} A_{3} a^{4} \overline{a}^{4}}{4 \omega_{0}^{6}} + \frac{165 A_{2} A_{5} a^{4} \overline{a}^{4}}{2 \omega_{0}^{4}} + \frac{121 A_{3} A_{4} a^{4} \overline{a}^{4}}{\omega_{0}^{4}} - \frac{70 A_{7} a^{4} \overline{a}^{4}}{\omega_{0}^{2}}$

$f_{2}=\frac{1151545 A_{1}^{7} a^{5} \overline{a}^{3}}{34992 \omega_{0}^{14}} - \frac{476867 A_{1}^{5} A_{2} a^{5} \overline{a}^{3}}{2592 \omega_{0}^{12}} + \frac{3947 A_{1}^{4} A_{3} a^{5} \overline{a}^{3}}{24 \omega_{0}^{10}} + \frac{453935 A_{1}^{3} A_{2}^{2} a^{5} \overline{a}^{3}}{1728 \omega_{0}^{10}} - \frac{96709 A_{1}^{3} A_{4} a^{5} \overline{a}^{3}}{720 \omega_{0}^{8}} - \frac{3482329 A_{1}^{2} A_{2} A_{3} a^{5} \overline{a}^{3}}{10800 \omega_{0}^{8}} + \frac{1041 A_{1}^{2} A_{5} a^{5} \overline{a}^{3}}{10 \omega_{0}^{6}} - \frac{26719 A_{1} A_{2}^{3} a^{5} \overline{a}^{3}}{384 \omega_{0}^{8}} + \frac{194993 A_{1} A_{2} A_{4} a^{5} \overline{a}^{3}}{1440 \omega_{0}^{6}} + \frac{10993 A_{1} A_{3}^{2} a^{5} \overline{a}^{3}}{135 \omega_{0}^{6}} - \frac{763 A_{1} A_{6} a^{5} \overline{a}^{3}}{12 \omega_{0}^{4}} + \frac{48401 A_{2}^{2} A_{3} a^{5} \overline{a}^{3}}{1200 \omega_{0}^{6}} - \frac{537 A_{2} A_{5} a^{5} \overline{a}^{3}}{20 \omega_{0}^{4}} - \frac{353 A_{3} A_{4} a^{5} \overline{a}^{3}}{9 \omega_{0}^{4}} + \frac{56 A_{7} a^{5} \overline{a}^{3}}{3 \omega_{0}^{2}},\qquad f_{-2}=\frac{1151545 A_{1}^{7} a^{3} \overline{a}^{5}}{34992 \omega_{0}^{14}} - \frac{476867 A_{1}^{5} A_{2} a^{3} \overline{a}^{5}}{2592 \omega_{0}^{12}} + \frac{3947 A_{1}^{4} A_{3} a^{3} \overline{a}^{5}}{24 \omega_{0}^{10}} + \frac{453935 A_{1}^{3} A_{2}^{2} a^{3} \overline{a}^{5}}{1728 \omega_{0}^{10}} - \frac{96709 A_{1}^{3} A_{4} a^{3} \overline{a}^{5}}{720 \omega_{0}^{8}} - \frac{3482329 A_{1}^{2} A_{2} A_{3} a^{3} \overline{a}^{5}}{10800 \omega_{0}^{8}} + \frac{1041 A_{1}^{2} A_{5} a^{3} \overline{a}^{5}}{10 \omega_{0}^{6}} - \frac{26719 A_{1} A_{2}^{3} a^{3} \overline{a}^{5}}{384 \omega_{0}^{8}} + \frac{194993 A_{1} A_{2} A_{4} a^{3} \overline{a}^{5}}{1440 \omega_{0}^{6}} + \frac{10993 A_{1} A_{3}^{2} a^{3} \overline{a}^{5}}{135 \omega_{0}^{6}} - \frac{763 A_{1} A_{6} a^{3} \overline{a}^{5}}{12 \omega_{0}^{4}} + \frac{48401 A_{2}^{2} A_{3} a^{3} \overline{a}^{5}}{1200 \omega_{0}^{6}} - \frac{537 A_{2} A_{5} a^{3} \overline{a}^{5}}{20 \omega_{0}^{4}} - \frac{353 A_{3} A_{4} a^{3} \overline{a}^{5}}{9 \omega_{0}^{4}} + \frac{56 A_{7} a^{3} \overline{a}^{5}}{3 \omega_{0}^{2}}$

$f_{4}=\frac{54067 A_{1}^{7} a^{6} \overline{a}^{2}}{34992 \omega_{0}^{14}} - \frac{283 A_{1}^{5} A_{2} a^{6} \overline{a}^{2}}{96 \omega_{0}^{12}} + \frac{3649 A_{1}^{4} A_{3} a^{6} \overline{a}^{2}}{3240 \omega_{0}^{10}} - \frac{3457 A_{1}^{3} A_{2}^{2} a^{6} \overline{a}^{2}}{576 \omega_{0}^{10}} + \frac{A_{1}^{3} A_{4} a^{6} \overline{a}^{2}}{81 \omega_{0}^{8}} + \frac{9529 A_{1}^{2} A_{2} A_{3} a^{6} \overline{a}^{2}}{1080 \omega_{0}^{8}} + \frac{61 A_{1}^{2} A_{5} a^{6} \overline{a}^{2}}{63 \omega_{0}^{6}} + \frac{2299 A_{1} A_{2}^{3} a^{6} \overline{a}^{2}}{384 \omega_{0}^{8}} - \frac{47 A_{1} A_{2} A_{4} a^{6} \overline{a}^{2}}{10 \omega_{0}^{6}} - \frac{878 A_{1} A_{3}^{2} a^{6} \overline{a}^{2}}{675 \omega_{0}^{6}} - \frac{49 A_{1} A_{6} a^{6} \overline{a}^{2}}{18 \omega_{0}^{4}} - \frac{2099 A_{2}^{2} A_{3} a^{6} \overline{a}^{2}}{800 \omega_{0}^{6}} + \frac{27 A_{2} A_{5} a^{6} \overline{a}^{2}}{70 \omega_{0}^{4}} - \frac{4 A_{3} A_{4} a^{6} \overline{a}^{2}}{15 \omega_{0}^{4}} + \frac{28 A_{7} a^{6} \overline{a}^{2}}{15 \omega_{0}^{2}},\qquad f_{-4}=\frac{54067 A_{1}^{7} a^{2} \overline{a}^{6}}{34992 \omega_{0}^{14}} - \frac{283 A_{1}^{5} A_{2} a^{2} \overline{a}^{6}}{96 \omega_{0}^{12}} + \frac{3649 A_{1}^{4} A_{3} a^{2} \overline{a}^{6}}{3240 \omega_{0}^{10}} - \frac{3457 A_{1}^{3} A_{2}^{2} a^{2} \overline{a}^{6}}{576 \omega_{0}^{10}} + \frac{A_{1}^{3} A_{4} a^{2} \overline{a}^{6}}{81 \omega_{0}^{8}} + \frac{9529 A_{1}^{2} A_{2} A_{3} a^{2} \overline{a}^{6}}{1080 \omega_{0}^{8}} + \frac{61 A_{1}^{2} A_{5} a^{2} \overline{a}^{6}}{63 \omega_{0}^{6}} + \frac{2299 A_{1} A_{2}^{3} a^{2} \overline{a}^{6}}{384 \omega_{0}^{8}} - \frac{47 A_{1} A_{2} A_{4} a^{2} \overline{a}^{6}}{10 \omega_{0}^{6}} - \frac{878 A_{1} A_{3}^{2} a^{2} \overline{a}^{6}}{675 \omega_{0}^{6}} - \frac{49 A_{1} A_{6} a^{2} \overline{a}^{6}}{18 \omega_{0}^{4}} - \frac{2099 A_{2}^{2} A_{3} a^{2} \overline{a}^{6}}{800 \omega_{0}^{6}} + \frac{27 A_{2} A_{5} a^{2} \overline{a}^{6}}{70 \omega_{0}^{4}} - \frac{4 A_{3} A_{4} a^{2} \overline{a}^{6}}{15 \omega_{0}^{4}} + \frac{28 A_{7} a^{2} \overline{a}^{6}}{15 \omega_{0}^{2}}$

$f_{6}=\frac{11 A_{1}^{7} a^{7} \overline{a}}{864 \omega_{0}^{14}} + \frac{211 A_{1}^{5} A_{2} a^{7} \overline{a}}{1728 \omega_{0}^{12}} + \frac{4 A_{1}^{4} A_{3} a^{7} \overline{a}}{15 \omega_{0}^{10}} - \frac{173 A_{1}^{3} A_{2}^{2} a^{7} \overline{a}}{1152 \omega_{0}^{10}} + \frac{23 A_{1}^{3} A_{4} a^{7} \overline{a}}{216 \omega_{0}^{8}} - \frac{13 A_{1}^{2} A_{2} A_{3} a^{7} \overline{a}}{24 \omega_{0}^{8}} - \frac{61 A_{1}^{2} A_{5} a^{7} \overline{a}}{294 \omega_{0}^{6}} - \frac{53 A_{1} A_{2}^{3} a^{7} \overline{a}}{256 \omega_{0}^{8}} - \frac{869 A_{1} A_{2} A_{4} a^{7} \overline{a}}{1680 \omega_{0}^{6}} + \frac{51 A_{1} A_{3}^{2} a^{7} \overline{a}}{350 \omega_{0}^{6}} + \frac{A_{1} A_{6} a^{7} \overline{a}}{56 \omega_{0}^{4}} - \frac{111 A_{2}^{2} A_{3} a^{7} \overline{a}}{2800 \omega_{0}^{6}} + \frac{57 A_{2} A_{5} a^{7} \overline{a}}{980 \omega_{0}^{4}} + \frac{11 A_{3} A_{4} a^{7} \overline{a}}{35 \omega_{0}^{4}} + \frac{8 A_{7} a^{7} \overline{a}}{35 \omega_{0}^{2}},\qquad f_{-6}=\frac{11 A_{1}^{7} a \overline{a}^{7}}{864 \omega_{0}^{14}} + \frac{211 A_{1}^{5} A_{2} a \overline{a}^{7}}{1728 \omega_{0}^{12}} + \frac{4 A_{1}^{4} A_{3} a \overline{a}^{7}}{15 \omega_{0}^{10}} - \frac{173 A_{1}^{3} A_{2}^{2} a \overline{a}^{7}}{1152 \omega_{0}^{10}} + \frac{23 A_{1}^{3} A_{4} a \overline{a}^{7}}{216 \omega_{0}^{8}} - \frac{13 A_{1}^{2} A_{2} A_{3} a \overline{a}^{7}}{24 \omega_{0}^{8}} - \frac{61 A_{1}^{2} A_{5} a \overline{a}^{7}}{294 \omega_{0}^{6}} - \frac{53 A_{1} A_{2}^{3} a \overline{a}^{7}}{256 \omega_{0}^{8}} - \frac{869 A_{1} A_{2} A_{4} a \overline{a}^{7}}{1680 \omega_{0}^{6}} + \frac{51 A_{1} A_{3}^{2} a \overline{a}^{7}}{350 \omega_{0}^{6}} + \frac{A_{1} A_{6} a \overline{a}^{7}}{56 \omega_{0}^{4}} - \frac{111 A_{2}^{2} A_{3} a \overline{a}^{7}}{2800 \omega_{0}^{6}} + \frac{57 A_{2} A_{5} a \overline{a}^{7}}{980 \omega_{0}^{4}} + \frac{11 A_{3} A_{4} a \overline{a}^{7}}{35 \omega_{0}^{4}} + \frac{8 A_{7} a \overline{a}^{7}}{35 \omega_{0}^{2}}$

$f_{8}=\frac{A_{1}^{7} a^{8}}{34992 \omega_{0}^{14}} + \frac{7 A_{1}^{5} A_{2} a^{8}}{7776 \omega_{0}^{12}} + \frac{49 A_{1}^{4} A_{3} a^{8}}{9720 \omega_{0}^{10}} + \frac{7 A_{1}^{3} A_{2}^{2} a^{8}}{1728 \omega_{0}^{10}} + \frac{29 A_{1}^{3} A_{4} a^{8}}{1620 \omega_{0}^{8}} + \frac{97 A_{1}^{2} A_{2} A_{3} a^{8}}{5400 \omega_{0}^{8}} + \frac{53 A_{1}^{2} A_{5} a^{8}}{1470 \omega_{0}^{6}} + \frac{A_{1} A_{2}^{3} a^{8}}{384 \omega_{0}^{8}} + \frac{61 A_{1} A_{2} A_{4} a^{8}}{2520 \omega_{0}^{6}} + \frac{79 A_{1} A_{3}^{2} a^{8}}{9450 \omega_{0}^{6}} + \frac{19 A_{1} A_{6} a^{8}}{504 \omega_{0}^{4}} + \frac{71 A_{2}^{2} A_{3} a^{8}}{16800 \omega_{0}^{6}} + \frac{13 A_{2} A_{5} a^{8}}{980 \omega_{0}^{4}} + \frac{A_{3} A_{4} a^{8}}{126 \omega_{0}^{4}} + \frac{A_{7} a^{8}}{63 \omega_{0}^{2}},\qquad f_{-8}=\frac{A_{1}^{7} \overline{a}^{8}}{34992 \omega_{0}^{14}} + \frac{7 A_{1}^{5} A_{2} \overline{a}^{8}}{7776 \omega_{0}^{12}} + \frac{49 A_{1}^{4} A_{3} \overline{a}^{8}}{9720 \omega_{0}^{10}} + \frac{7 A_{1}^{3} A_{2}^{2} \overline{a}^{8}}{1728 \omega_{0}^{10}} + \frac{29 A_{1}^{3} A_{4} \overline{a}^{8}}{1620 \omega_{0}^{8}} + \frac{97 A_{1}^{2} A_{2} A_{3} \overline{a}^{8}}{5400 \omega_{0}^{8}} + \frac{53 A_{1}^{2} A_{5} \overline{a}^{8}}{1470 \omega_{0}^{6}} + \frac{A_{1} A_{2}^{3} \overline{a}^{8}}{384 \omega_{0}^{8}} + \frac{61 A_{1} A_{2} A_{4} \overline{a}^{8}}{2520 \omega_{0}^{6}} + \frac{79 A_{1} A_{3}^{2} \overline{a}^{8}}{9450 \omega_{0}^{6}} + \frac{19 A_{1} A_{6} \overline{a}^{8}}{504 \omega_{0}^{4}} + \frac{71 A_{2}^{2} A_{3} \overline{a}^{8}}{16800 \omega_{0}^{6}} + \frac{13 A_{2} A_{5} \overline{a}^{8}}{980 \omega_{0}^{4}} + \frac{A_{3} A_{4} \overline{a}^{8}}{126 \omega_{0}^{4}} + \frac{A_{7} \overline{a}^{8}}{63 \omega_{0}^{2}}$

第二阶计算(以下为手动计算，有错误,请勿参考）：

$$ \omega_2 = \frac{2A_1(f^{(1)}*f^{(0)})_{\pm 1}+A_2(f^{(0)})^3_{\pm 1}}{2\omega_0 f^{(0)}_{\pm 1}}=\frac{5A_1^2}{3\omega_0^3}(a^2+b^2)\\ $$$$ f^0_{\pm 1}=a\pm i b\\ (f^0)^3_{\pm 1}=0\\ (f^{1}*f^{0})_{\pm 1}=-\frac{5A_1(a^2+b^2)(a\pm i b)}{3\omega_0^2} $$

代入$\omega_2$的值：

# show(dconv(res_conv_f[1][0],res_conv_f[0][0]))
ret=dconv(dconv(res_f[0],res_f[0]),res_f[0])
show(ret)
#计算卷积的结果（a,b）：
conv=[[0 for _ in range(-MAXLEN,MAXLEN+1)] for i in range(iteration)]
dconv(res_f[0],res_f[0],conv[0]) #conv[i]为i+1次卷积得到的结果
for i in range(iteration-1):
    dconv(res_f[0],conv[i],conv[i+1])
#把最高阶的项表示出来，相当于求ax+b=0,这样当我们有了低阶项之后 可以一项一项地向上计算

fn_res=[]
# display(Markdown("# 求出各个f^i的表达式："))
for i in range(iteration):
    result = sympy.solve(coeffs[i],f_sym[i])[0]
    fn_res.append(result)
    # display(Latex(f"${sympy.latex(f_sym[i])}={sympy.latex(fn_res[i])}$"))
omega_res=[None]
#
# for i in range(1,iteration):# 第零阶不可计算
#     coeffs_1=coeffs[i].subs(n,1)# 代入n=1，计算omega的值
#     omega_res.append(sympy.solve(coeffs_1,omegas[i])[0])
    # display(Latex(f"${sympy.latex(omegas[i])}={sympy.latex(omega_res[i])}$")) 
#我们通过第一阶的计算，求出了omega_1=0，和$f_n^(1)$的值(n=+-1 undefined?)。用expr.subs来化简式子：

for i in range(1,iteration):
    coeffs_1 = coeffs[i].subs(n,1)
    omega_res.append(sympy.expand(sympy.solve(coeffs_1,omegas[i])[0]))
    display(Latex(f"${sympy.latex(omegas[i])}={sympy.latex(omega_res[i])}$")) 

# display(Markdown("# 代入$\omega_1=0$之后："))
res_f[1] = [params[1]/params[0]/(n**2-1) *dconv_n(res_f[0],res_f[0],n) if abs(n)!=1 else 0  for n in range(-MAXLEN,MAXLEN+1)]

for i in range(iteration):
    fn_res[i]=fn_res[i].subs(omegas[1],0) #化简
    # display(Latex(f"${sympy.latex( f_sym[i])}={sympy.latex(fn_res[i])}$"))
show(res_f[1])
for i in range(iteration):
    fn_res[i]=fn_res[i].subs(omegas[2],5*params[1]**2/3/omegas[0]**3*(a**2+b**2))
    # display(Latex(f"${sympy.latex(f_sym[i])}={sympy.latex(fn_res[i])}$"))

res_f[2]=[(2*params[1] *dconv_n(res_f[0],res_f[1],n)+ params[2] *conv[1][getidx(n)] -10*params[1]**2/(3*omegas[0]**2) *n**2*res_f[0][getidx(n)]*(a**2+b**2) )/((n**2-1)*params[0]) if abs(n)!=1 else 0 for n in range(-MAXLEN,MAXLEN+1)]
# show(coef[0],0)
# show(res_f[1],1)
show(res_f[2],2)

$\displaystyle \overline{a}$

第三阶计算：

$$ \omega_3=0 $$

# show(conv[2]) +-1项为0
# show(dconv(res_f[1],res_f[1])) #+-1项为0
# show(dconv(res_f[0],res_f[2])) #0
# show(dconv(conv[0],res_f[1])) #0
# show(res_f[1]) #0
# show(res_f[0]) #not 0
for i in range(iteration):
    fn_res[i]=fn_res[i].subs(omegas[3],0)
    # display(Latex(f"${sympy.latex(f_sym[i])}={sympy.latex(fn_res[i])}$"))

res_f[3]=[(-(10*params[1]**2*res_f[1][getidx(n)]*n**2*(a**2+b**2))/(3*params[0])+2*params[1]*dconv_n(res_f[0],res_f[2],n)+params[1]*dconv_n(res_f[1],res_f[1],n)+3*params[2]*dconv_n(conv[0],res_f[1],n)+params[3]*conv[2][getidx(n)] )/(params[0]*(n**2-1)) if abs(n)!=1 else 0 for n in range(-MAXLEN,MAXLEN+1)]
show(res_f[3])

$f_0=- \frac{\frac{38 A_{1}^{3} \left(a - 1.0 i b\right)^{2} \left(a + 1.0 i b\right)^{2}}{9 \omega_{0}^{4}} - \frac{10 A_{1} A_{2} \left(a - 1.0 i b\right)^{2} \left(a + 1.0 i b\right)^{2}}{\omega_{0}^{2}} + 6 A_{3} \left(a - 1.0 i b\right)^{2} \left(a + 1.0 i b\right)^{2}}{\omega_{0}^{2}}$

$f_{1}=0,\qquad f_{-1}=0$

$f_{2}=\frac{- \frac{4 A_{1}^{3} \left(a - 1.0 i b\right) \left(a + 1.0 i b\right)^{3}}{3 \omega_{0}^{4}} - \frac{40 A_{1}^{3} \left(a + 1.0 i b\right)^{2} \left(a^{2} + b^{2}\right)}{9 \omega_{0}^{4}} - \frac{4 A_{1} A_{2} \left(a - 1.0 i b\right) \left(a + 1.0 i b\right)^{3}}{\omega_{0}^{2}} - \frac{A_{1} \left(a - 1.0 i b\right) \left(- \frac{2 A_{1}^{2} \left(a + 1.0 i b\right)^{3}}{3 \omega_{0}^{2}} - A_{2} \left(a + 1.0 i b\right)^{3}\right)}{4 \omega_{0}^{2}} + 4 A_{3} \left(a - 1.0 i b\right) \left(a + 1.0 i b\right)^{3}}{3 \omega_{0}^{2}},\qquad f_{-2}=\frac{- \frac{4 A_{1}^{3} \left(a - 1.0 i b\right)^{3} \left(a + 1.0 i b\right)}{3 \omega_{0}^{4}} - \frac{40 A_{1}^{3} \left(a - 1.0 i b\right)^{2} \left(a^{2} + b^{2}\right)}{9 \omega_{0}^{4}} - \frac{4 A_{1} A_{2} \left(a - 1.0 i b\right)^{3} \left(a + 1.0 i b\right)}{\omega_{0}^{2}} - \frac{A_{1} \left(a + 1.0 i b\right) \left(- \frac{2 A_{1}^{2} \left(a - 1.0 i b\right)^{3}}{3 \omega_{0}^{2}} - A_{2} \left(a - 1.0 i b\right)^{3}\right)}{4 \omega_{0}^{2}} + 4 A_{3} \left(a - 1.0 i b\right)^{3} \left(a + 1.0 i b\right)}{3 \omega_{0}^{2}}$

$f_{3}=0,\qquad f_{-3}=0$

$f_{4}=\frac{\frac{A_{1}^{3} \left(a + 1.0 i b\right)^{4}}{9 \omega_{0}^{4}} + \frac{A_{1} A_{2} \left(a + 1.0 i b\right)^{4}}{\omega_{0}^{2}} - \frac{A_{1} \left(a + 1.0 i b\right) \left(- \frac{2 A_{1}^{2} \left(a + 1.0 i b\right)^{3}}{3 \omega_{0}^{2}} - A_{2} \left(a + 1.0 i b\right)^{3}\right)}{4 \omega_{0}^{2}} + A_{3} \left(a + 1.0 i b\right)^{4}}{15 \omega_{0}^{2}},\qquad f_{-4}=\frac{\frac{A_{1}^{3} \left(a - 1.0 i b\right)^{4}}{9 \omega_{0}^{4}} + \frac{A_{1} A_{2} \left(a - 1.0 i b\right)^{4}}{\omega_{0}^{2}} - \frac{A_{1} \left(a - 1.0 i b\right) \left(- \frac{2 A_{1}^{2} \left(a - 1.0 i b\right)^{3}}{3 \omega_{0}^{2}} - A_{2} \left(a - 1.0 i b\right)^{3}\right)}{4 \omega_{0}^{2}} + A_{3} \left(a - 1.0 i b\right)^{4}}{15 \omega_{0}^{2}}$

$f_{5}=0,\qquad f_{-5}=0$

$f_{6}=0,\qquad f_{-6}=0$

$f_{7}=0,\qquad f_{-7}=0$

$f_{8}=0,\qquad f_{-8}=0$

$f_{9}=0,\qquad f_{-9}=0$

$f_{10}=0,\qquad f_{-10}=0$

u,v=sympy.symbols("u v ")
w=sympy.expand(u*(v+1))
w=(u*(v+1))
sympy.srepr(w)
sympy.Pow(w,2)
expr = sympy.Add(u,u)
expr.func
sympy.core.numbers.Zero()
expr = 3*u **2 * v
expr
expr.func,expr.args
w=expr.args[2]
w.args[0].args