茶年 · 中文维基百科

马丢函数

Wikipedia · 2025-12-02 维基百科 👁 1 阅读

马丟函数（）是1868年法國數學家因研究数学物理所推得的特殊函數，下列马丟方程的解析解：

: \frac{d^2y}{dx^2}+[a-2q\cos (2x) ]y=0.

马丟方程有两个线性无关的解：
;奇数解
MathieuCE(n, q, x)，或记为w_{I}(n,q,x),

;偶数解
MathieuSE(n, q, x).或记为w_{II}(n,q,x)
称为基本解

周期性
马丟函数 MathieuC(a,q,z) 或 MathieuS(a,q,z) 只有一个是周期为 \pi 或2\pi的周期解，另一个不是。

马丟函数 MathieuC(a,q,z) 和 MathieuS(a,q,z) 两者都有是周期为2n\pi（n≥2)的周期函数。

File:MathieuCE 2d 1.gif
File:MathieuCE 2d 2.gif
File:MathieuCE 2d 4.gif
File:MathieuSE 2d 3.gif

正交性

  • \int_{0}^{2\pi}\! ce_m(x,q)ce_n(x,q)\,dx=0
  • \int_{0}^{2\pi}\! ce_m(x,q)se_n(x,q)\,dx=0
  • \int_{0}^{2\pi}\! se_m(x,q)se_n(x,q)\,dx=0

特征方程
马丟方程的特征方程是

cos(\pi*v)=w_I(a,q,\pi)

cos(\pi*v)=w_{II}(b,q,\pi)

对于给定的v，q, 上列特征方程给出无穷多个a、b解称为特征值。

特征值的展开
马丟函数体特征值可展开成级数：

a_0(q)= {-(1/2)z^2+(7/128)z^4-(29/2304)z^6+(68687/18874368)z^8+\mathcal{O}(z^{10})}
a_1(q)= {1+z-(1/8)z^2-(1/64)z^3-(1/1536)z^4+(11/36864)z^5+(49/589824)z^6+(55/9437184)z^7-(83/35389440)z^8-(12121/15099494400)z^9+\mathcal{O}(z^{10})}
a_2(q)={4+(5/12)z^2-(763/13824)z^4+(1002401/79626240)z^6-(1669068401/458647142400)z^8+\mathcal{O}(z^{10})}
a_3(q)= {9+(1/16)z^2+(1/64)z^3+(13/20480)z^4-(5/16384)z^5-(1961/23592960)z^6-(609/104857600)z^7+(4957199/2113929216000)z^8+(872713/1087163596800)z^9+\mathcal{O}(z^{10})}

b_1(q)={1-z-(1/8)z^2+(1/64)z^3-(1/1536)z^4-(11/36864)z^5+(49/589824)z^6-(55/9437184)z^7-(83/35389440)z^8+(12121/15099494400)z^9+\mathcal{O}(z^{10})}
b_2(q)={4-(1/12)z^2+(5/13824)z^4-(289/79626240)z^6+(21391/458647142400)z^8+\mathcal{O}(z^{10})}
b_3(q)= {9+(1/16)z^2-(1/64)z^3+(13/20480)z^4+(5/16384)z^5-(1961/23592960)z^6+(609/104857600)z^7+(4957199/2113929216000)z^8-(872713/1087163596800)z^9+\mathcal{O}(z^{10})}
b_4(q)= {16+(1/30)z^2-(317/864000)z^4+(10049/2721600000)z^6-(93824197/2006581248000000)z^8+\mathcal{O}(z^{10})}
b_5(q)= {25+(1/48)z^2+(11/774144)z^4-(1/147456)z^5+(37/891813888)z^6+(7/339738624)z^7+(63439/201364441399296)z^8+(1/2130840649728)*z^9+\mathcal{O}(z^{10})}

级数展开
马丟函数ce,se的级数展开

ce_0(z,q)= {1-(1/2)cos(2z)q+(-1/16+(1/32)cos(4z))q^2+((11/128)cos(2z)-(1/1152)cos(6z))*q^3+O(q^4)}
ce_1(z,q)= {cos(z)-(1/8)cos(3z)q+(-(1/128)cos(z)-(1/64)cos(3z)+(1/192)cos(5z))q^2+(-(1/512)cos(z)+(1/3072)cos(3z)+(1/1152)cos(5z)-(1/9216)cos(7z))*q^3+O(q^4)}
ce_2(z,q)= {cos(2z)+(1/4-(1/12)cos(4z))q+(-(19/288)cos(2z)+(1/384)cos(6z))q^2+(-49/1152+(11/4608)cos(4z)-(1/23040)cos(8z))q^3+O(q^4)}
ce_3(z,q)= {cos(3z)+((1/8)cos(z)-(1/16)cos(5z))q+(-(5/512)cos(3z)+(1/64)cos(z)+(1/640)cos(7z))q^2+(-(1/512)cos(3z)-(1/4096)cos(z)+(11/40960)cos(5z)-(1/46080)cos(9z))*q^3+O(q^4)}
ce_4(z,q)= {cos(4z)+((1/12)cos(2z)-(1/20)cos(6z))q+(-(17/3600)cos(4z)+1/192+(1/960)cos(8z))q^2+((7/28800)cos(2z)+(29/288000)cos(6z)-(1/80640)cos(10z))q^3+O(q^4)}

se_1(z,q)= {sin(z)-(1/8)sin(3z)q+(-(1/128)sin(z)+(1/64)sin(3z)+(1/192)sin(5z))q^2+((1/512)sin(z)+(1/3072)sin(3z)-(1/1152)sin(5z)-(1/9216)sin(7z))*q^3+O(q^4)}
se_2(z,q)= {sin(2z)-(1/12)sin(4z)q+(-(1/288)sin(2z)+(1/384)sin(6z))q^2+((1/1536)sin(4z)-(1/23040)sin(8z))q^3+O(q^4)}
se_3(z,q)= {sin(3z)+((1/8)sin(z)-(1/16)sin(5z))q+(-(5/512)sin(3z)-(1/64)sin(z)+(1/640)sin(7z))q^2+((1/512)sin(3z)-(1/4096)sin(z)+(11/40960)sin(5z)-(1/46080)sin(9z))*q^3+O(q^4)}
se_4(z,q)= {sin(4z)+((1/12)sin(2z)-(1/20)sin(6z))q+(-(17/3600)sin(4z)+(1/960)sin(8z))q^2+(-(1/1600)sin(2z)+(29/288000)sin(6z)-(1/80640)sin(10z))q^3+O(q^4)}
se_5(z,q)= {sin(5z)+((1/16)sin(3z)-(1/24)sin(7z))q+(-(13/4608)sin(5z)+(1/384)sin(z)+(1/1344)sin(9z))q^2+(-(7/73728)sin(3z)+(13/258048)sin(7z)-(1/9216)sin(z)-(1/129024)sin(11z))q^3+O(q^4)}

傅立叶展开式
马丟函数的傅立叶展开：

  • MathieuCE(2n, q, x)=\sum_{m=0}^{\infty}A_{2m}^{2n}(q)cos(2mx)
  • MathieuCE(2n+1, q, x)=\sum_{m=0}^{\infty}A_{2m+1}^{2n+1}(q)cos[(2m+1)x]

*MathieuSE(2n+1,q,x)=\sum_{m=0}^{\infty}B_{2m+1}^{2n+1}(q)sin[(2m+1)x]
*MathieuSE(2n+2,q,x)=\sum_{m=0}^{\infty}B_{2m+2}^{2n+2}(q)sin[(2m+2)x]

其中系数A,B满足下列递归关系：

aA_0=qA_2

(a-4)A_2=q(2A_0+A_4)

(a-4m^2)A_{2m}=q(A_{2m-2}+A_{2m+2})

(a-1+q)B_1=qB_3

(a-(2m+1)^2)B_{2m+1}=q(B_{2m-1}+B_{2m+1})

关系式
马丟方程的基本解W_I W_{II} 满足下列关系：:

:\begin{vmatrix} w_I(n,q,0) & w_{II}(n,q,0)\\w_{i}^'(n,q,0) & w_{II}^'(n,q,0) \end{vmatrix}= \begin{vmatrix} 1 & 0\\0 & 1 \end{vmatrix}

郎斯基行列式：
W[w_I,w_{II}]=1

w_I(a,q,z+\pi)=w_I(a,q,\pi)w_I(a,q,z)+w_I^'(a,q,\pi)w_{II}(a,q,z)
w_I(a,q,z-\pi)=w_I(a,q,\pi)w_I(a,q,z)-w_I^'(a,q,\pi)w_{II}(a,q,z)
w_{II}(a,q,z+\pi)=w_{II}(a,q,\pi)w_{II}(a,q,z)+w_{II}^'(a,q,\pi)w_{II}(a,q,z)
w_{II}(a,q,z-\pi)=w_{II}(a,q,\pi)w_{II}(a,q,z)-w_I^'(a,q,\pi)w_{II}(a,q,z)

w_I(-z)=w_I(z)
w_{II}(-z)=-w_{II}(z)

特例

  • CE(a,0,z)=cos(az)
  • SE(a,0,z)=sin(az)
  • MathieuA(1,0)=1
  • MathieuA(a,0)=a^2
  • MathieuB(a,0)=a^2
  • MathieuFloquet(a,0,z)=exp(Isqrt(a)z)

*

夫洛开解
马丟函数中，如果f(x) 是一个周期为\omega的解，并满足下列条件

f(x+\omega)=\sigma*f(x),其中\sigma与x 无关，则此解称为夫洛开解。

;级数展开
MF(1, 1, z) = {.7992-.5734I+(-.9134+.6553I)z+(.3996-.2867I)z^2+(-.1523+.1092I)z^3+(-.2331+.1673I)*z^4+O(z^5)}
MF(1, 2, z) = {.7643-.4526I+(-1.167+.6910I)z+(1.146-.6789I)z^2+(-.5835+.3455I)z^3+(-.2229+.1320I)*z^4+O(z^5)}
MF(1, 3, z) = {.6841-.3703I+(-1.318+.7135I)z+(1.710-.9258I)z^2+(-1.098+.5946I)z^3+(0.2851e-1-0.1543e-1I)*z^4+O(z^5)}

参考文献
*王竹溪 郭敦仁 《特殊函数概论》 第十二章 马丟函数 北京大学出版社 2000
*Frank J Oliver NIST Handbook of Mathematical Functions,Cambridge University PRESS, 2010

#特殊函数 #常微分方程

评论 (0)

  • 还没有评论，来抢沙发吧。