螺旋函数

**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

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

## 圖例

**螺旋葉數與鏡像對稱**

* * * * * * * * * * * *

**螺旋葉形**

* * * * * * *

## 參考文獻