查看“︁螺旋函数”︁的源代码

[[File:Swirl0.JPG|thumb|300px|swirl  function  Maple plot]]
'''螺旋函数'''(Swirl function)是一个以[[三角函数]]定义的[[特殊函数]]<ref>Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 36-37 and 86, 1999. </ref>：


<math>S(k,n,r,\theta)=sin(k*cos(r)-n*\theta)</math>

其中k,n均为整数。k与螺旋叶的长度与形状有关，n为螺旋的叶片数。

==对称性==
;镜像对称
*<math>S(k,n,r,\theta)</math>与<math>S(k,-n,r,\theta)</math> 互为镜像对称.
*<math>f(-k, n, r, \theta)=-f(k, n, r, -\theta)     </math>
*<math>f(-k, n, r, \theta)=-f(k, -n, r, \theta)     </math>
*<math>f(-k, -n, r, \theta)=-f(k, n, r, \theta)     </math>
*<math>f(-k, n, r, -\theta)=-f(k, n, r, \theta)                  </math>
*<math>  f(-k, n, r, \theta)=-f(k, n, -r, -\theta)                </math>

*<math> f(-k, n, -r, -\theta)=-f(k, n, r, \theta)                 </math>

*<math>   f(-k, -n, -r, \theta)=-f(k, n, r, \theta)               </math>

*<math> f(-k, n, -r, -\theta)=-f(k, n, r, \theta)                 </math>

;全对称
*<math>f(k, -n, r, \theta)=f(k, n, r, -\theta)</math>
*<math>f(k, -n, r, -\theta)=f(k, n, r, \theta)                  </math>

*<math> f(k, n, -r, \theta)=f(k, n, r, \theta)                 </math>

*<math> f(k, n, -r, \theta)=f(k, n, r, \theta)                 </math>

*<math> f(k, n, -r, \theta)=f(k, -n, r, -\theta)                 </math>

*<math>  f(k, -n, -r, -\theta)=f(k, n, r, \theta)           </math>
*<math> f(k, n, -r, \theta)-f(k, n, r, \theta)            </math>

;旋转对称
<math>S(k,n,r,\theta+\frac{2\pi}{n})=S(k,n,r,\theta)</math>

==级数展开==
<math>S(k,n,r,\theta) \approx {sin(k-n*\theta)-(1/2)*cos(k-n*\theta)*k*r^2+(-(1/8)*sin(k-n*\theta)*k^2+(1/24)*cos(k-n*\theta)*k)*r^4+((1/48)*sin(k-n*\theta)*k^2+cos(k-n*\theta)*(-(1/720)*k+(1/48)*k^3))*r^6+O(r^8)}</math>

<math>S(k,n,r,\theta) \approx {sin(k*cos(r))-cos(k*cos(r))*n*\theta-(1/2)*sin(k*cos(r))*n^2*\theta^2+(1/6)*cos(k*cos(r))*n^3*\theta^3+(1/24)*sin(k*cos(r))*n^4*\theta^4-(1/120)*cos(k*cos(r))*n^5*\theta^5-(1/720)*sin(k*cos(r))*n^6*\theta^6+(1/5040)*cos(k*cos(r))*n^7*\theta^7+(1/40320)*sin(k*cos(r))*n^8*\theta^8+O(\theta^9)}</math>

==与其他特殊函数关系==

*<math>S(k,n,r,\theta)={ \frac{ \left( nx\arccos \left( x \right) +1/2\,\pi  \right) 
{{\rm KummerM}\left(1,\,2,\,i \left( 2\,nx\arccos\left( x \right) +\pi  \right) \right)}
}{{{\rm e}^{1/2\,i \left( 2\,nx\arccos \left( x \right) +\pi  \right) 
}}}}</math>

*<math>S(k,n,r,\theta)={\frac {-i \left( 2\,nx\arccos \left( x \right) +\pi  \right) 
{{\rm WhittakerM}\left(0,\,1/2,\,i \left( 2\,nx\arccos \left( x \right) +\pi  \right) \right)}
}{4\,nx\arccos \left( x \right) +2\,\pi }}
</math>

*<math>S(k,n,r,\theta)={\frac {-1/2\,i \left( -1+{{\rm e}^{i \left( 2\,nx\arccos \left( x
 \right) +\pi  \right) }} \right) }{{{\rm e}^{1/2\,i \left( 2\,nx
\arccos \left( x \right) +\pi  \right) }}}}
</math>

*<math>S(k,n,r,\theta)=-n{x}^{2}{\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {1/2\,i \left( 2\,n
x \left( 1/2\,\pi -x{\it HeunC} \left( 0,1/2,0,0,1/4,{\frac {{x}^{2}}{
{x}^{2}-1}} \right) {\frac {1}{\sqrt {1-{x}^{2}}}} \right) +\pi 
 \right) } \right) {\it HeunC} \left( 0,1/2,0,0,1/4,{\frac {{x}^{2}}{{
x}^{2}-1}} \right) {\frac {1}{\sqrt {1-{x}^{2}}}} \left( {{\rm e}^{-1/
2\,i \left( -nx\pi \,\sqrt {1-{x}^{2}}+2\,n{x}^{2}{\it HeunC} \left( 0
,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}} \right) -\pi \,\sqrt {1-{x}^
{2}} \right) {\frac {1}{\sqrt {1-{x}^{2}}}}}} \right) ^{-1}+1/2\,\pi 
\, \left( nx+1 \right) {\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {1/2
\,i \left( 2\,nx \left( 1/2\,\pi -x{\it HeunC} \left( 0,1/2,0,0,1/4,{
\frac {{x}^{2}}{{x}^{2}-1}} \right) {\frac {1}{\sqrt {1-{x}^{2}}}}
 \right) +\pi  \right) } \right)  \left( {{\rm e}^{-1/2\,i \left( -nx
\pi \,\sqrt {1-{x}^{2}}+2\,n{x}^{2}{\it HeunC} \left( 0,1/2,0,0,1/4,{
\frac {{x}^{2}}{{x}^{2}-1}} \right) -\pi \,\sqrt {1-{x}^{2}} \right) {
\frac {1}{\sqrt {1-{x}^{2}}}}}} \right) ^{-1}
</math>

==图例==
;螺旋叶数与镜像对称
<gallery>
File:Swirl minus2.JPG|7,-2
File:Swirl2.JPG|7,2
File:Swirl minus4.JPG|7,-4
File:Swirl4.JPG|7,4
File:Swirl minus6.JPG|7,-6
File:Swirl6.JPG|7,6
File:Swirl minus8.JPG|7,-8
File:Swirl8.JPG|7,8
File:Swirl minus10.JPG|7,-10
File:Swirl10.JPG|7,10
File:Swirl minus12.JPG|7,-12
File:Swirl12.JPG|7,12
</gallery>
;螺旋叶形
<gallery>
File:Swirl04.JPG|0,4
File:Swirl14.JPG|1,4
File:Swirl24.JPG|2,4
File:Swirl74.JPG|7,4
File:Swirl-54.JPG|-5,4
File:Swirl-94.JPG|-9,4
File:Swirlk30.JPG|30,4
</gallery>

==参考文献==
<references/>

[[Category:特殊函数]]