螺旋函数

螺旋函数(Swirl function)是一个以三角函数定义的特殊函数：

S(k,n,r,\theta)=sin(kcos(r)-n\theta)

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
*S(k,n,r,\theta)与S(k,-n,r,\theta) 互为镜像对称.
*f(-k, n, r, \theta)=-f(k, n, r, -\theta)
*f(-k, n, r, \theta)=-f(k, -n, r, \theta)
*f(-k, -n, r, \theta)=-f(k, n, r, \theta)
*f(-k, n, r, -\theta)=-f(k, n, r, \theta)

f(-k, n, r, \theta)=-f(k, n, -r, -\theta)

f(-k, n, -r, -\theta)=-f(k, n, r, \theta)

f(-k, -n, -r, \theta)=-f(k, n, r, \theta)

f(-k, n, -r, -\theta)=-f(k, n, r, \theta)

;全对称
*f(k, -n, r, \theta)=f(k, n, r, -\theta)
*f(k, -n, r, -\theta)=f(k, n, r, \theta)

f(k, n, -r, \theta)=f(k, n, r, \theta)

f(k, n, -r, \theta)=f(k, n, r, \theta)

f(k, n, -r, \theta)=f(k, -n, r, -\theta)

f(k, -n, -r, -\theta)=f(k, n, r, \theta)
f(k, n, -r, \theta)-f(k, n, r, \theta)

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

级数展开
S(k,n,r,\theta) \approx {sin(k-n\theta)-(1/2)cos(k-n\theta)kr^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)}

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

与其他特殊函数关系
*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)
}}}}

*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 }}

*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) }}}}

*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}

图例
;螺旋叶数与镜像对称

;螺旋叶形

参考文献

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