符号函数的Taylor级数展开式

森之张卫东

符号函数的Taylor级数展开式
函数  taylor
格式  r = taylor(f,n,v)   %返回符号表达式f中的、指定的符号自变量v（若表达式f中有多个变量时）的n-1阶的Maclaurin多项式（即在零点附近v=0）近似式，其中v可以是字符串或符号变量。
r = taylor(f)      %返回符号表达式f中的、符号变量v的6阶的Maclaurin多项式（即在零点附近v=0）近似式，其中v=findsym(f)。
r = taylor(f,n,v,a)   %返回符号表达式f中的、指定的符号自变量v的n-1阶的Taylor级数（在指定的a点附近v=a）的展开式。其中a可以是一数值、符号、代表一数字值的字符串或未知变量。我们指出的是，用户可以以任意的次序输入参量n、v与a，命令taylor能从它们的位置与类型确定它们的目的。解析函数f(x)在点x=a的Taylor级数定义为：
例3-46
>>syms x y a pi m m1 m2
>>f = sin(x+pi/3);
>>T1 = taylor(f)
>>T2 = taylor(f,9)
>>T3 = taylor(f,a)
>>T4 = taylor(f,m1,m2)
>>T5 = taylor(f,m,a)
>>T6 = taylor(f,y)
>>T7 = taylor(f,y,m)   % 或taylor(f,m,y)
>>T8 = taylor(f,m,y,a)
>>T9 = taylor(f,y,a)
计算结果为：
T1 =
1/2*3^(1/2)+1/2*x-1/4*3^(1/2)*x^2-1/12*x^3+1/48*3^(1/2)*x^4+1/240*x^5
T2 =
1/2*3^(1/2)+1/2*x-1/4*3^(1/2)*x^2-1/12*x^3+1/48*3^(1/2)*x^4+1/240*x^5-1/1440*3^(1/2)* x^6-1/10080*x^7+1/80640*3^(1/2)*x^8
T3 =
sin(a+1/3*pi)+cos(a+1/3*pi)*(x-a)-1/2*sin(a+1/3*pi)*(x-a)^2-1/6*cos(a+1/3*pi)*  (x-a)^3+1/24*sin(a+1/3*pi)*(x-a)^4+1/120*cos(a+1/3*pi)*(x-a)^5
T4 =
sin(m2+1/3*pi)+cos(m2+1/3*pi)*(x-m2)-1/2*sin(m2+1/3*pi)*(x-m2)^2-1/6* cos(m2+1/3*pi)*(x-m2)^3+1/24*sin(m2+1/3*pi)*(x-m2)^4+1/120*
cos(m2+1/3*pi)*(x-m2)^5
T5 =
sin(a+1/3*pi)+cos(a+1/3*pi)*(x-a)-1/2*sin(a+1/3*pi)*(x-a)^2-1/6*cos(a+1/3*pi)*  (x-a)^3+1/24*sin(a+1/3*pi)*(x-a)^4+1/120*cos(a+1/3*pi)*(x-a)^5
T6 =
sin(y+1/3*pi)+cos(y+1/3*pi)*(x-y)-1/2*sin(y+1/3*pi)*(x-y)^2-1/6*cos(y+1/3*pi) *(x-y)^3+1/24*sin(y+1/3*pi)*(x-y)^4+1/120*cos(y+1/3*pi)*(x-y)^5
T7 =
sin(m+1/3*pi)+cos(m+1/3*pi)*(x-m)-1/2*sin(m+1/3*pi)*(x-m)^2-1/6*cos(m+1/3*pi) *(x-m)^3+1/24*sin(m+1/3*pi)*(x-m)^4+1/120*cos(m+1/3*pi)*(x-m)^5
T8 =
sin(a+1/3*pi)+cos(a+1/3*pi)*(x-a)-1/2*sin(a+1/3*pi)*(x-a)^2-1/6*cos(a+1/3*pi)*  (x-a)^3+1/24*sin(a+1/3*pi)*(x-a)^4+1/120*cos(a+1/3*pi)*(x-a)^5
T9 =
sin(a+1/3*pi)+cos(a+1/3*pi)*(x-a)-1/2*sin(a+1/3*pi)*(x-a)^2-1/6*cos(a+1/3*pi)*  (x-a)^3+1/24*sin(a+1/3*pi)*(x-a)^4+1/120*cos(a+1/3*pi)*(x-a)^5