螺旋函數

螺旋函數(Swirl function)是一個以三角函數定義的特殊函數：

$$S(k,n,r,\theta)=sin(k*cos(r)-n*\theta)$$

其中k,n均為整數。k與螺旋葉的長度與形狀有關，n為螺旋的葉片數。

## 對稱性

鏡像對稱

- $S(k,n,r,\theta)$與$S(k,-n,r,\theta)$ 互為鏡像對稱.
- $$f(-k, n, r, \theta)=-f(k, n, r, -\theta)$$
- $$f(-k, n, r, \theta)=-f(k, -n, r, \theta)$$
- $$f(-k, -n, r, \theta)=-f(k, n, r, \theta)$$
- $$f(-k, n, r, -\theta)=-f(k, n, r, \theta)$$
- $$f(-k, n, r, \theta)=-f(k, n, -r, -\theta)$$
- $$f(-k, n, -r, -\theta)=-f(k, n, r, \theta)$$
- $$f(-k, -n, -r, \theta)=-f(k, n, r, \theta)$$
- $$f(-k, n, -r, -\theta)=-f(k, n, r, \theta)$$

全對稱

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

旋轉對稱

$$S(k,n,r,\theta+\frac{2\pi}{n})=S(k,n,r,\theta)$$

## 級數展開

$$S(k,n,r,\theta) \approx {sin(k-n*\theta)-(1/2)*cos(k-n*\theta)*k*r^2+(-(1/8)*sin(k-n*\theta)*k^2+(1/24)*cos(k-n*\theta)*k)*r^4+((1/48)*sin(k-n*\theta)*k^2+cos(k-n*\theta)*(-(1/720)*k+(1/48)*k^3))*r^6+O(r^8)}$$

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

## 與其他特殊函數關係

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

## 圖例

螺旋葉數與鏡像對稱

- 7,-2
- 7,2
- 7,-4
- 7,4
- 7,-6
- 7,6
- 7,-8
- 7,8
- 7,-10
- 7,10
- 7,-12
- 7,12

螺旋葉形

- 0,4
- 1,4
- 2,4
- 7,4
- -5,4
- -9,4
- 30,4