斐波那契双曲函数

斐波那契双曲函数（Fibonoacci hyperbolic functions）是一个与黄金分割有关的特殊函数。

斐波那契双曲函数
定义如下：

;斐波那契双曲正弦函数
sFh(x)=\frac{ 2sinh(2x*\alpha)}{\sqrt{5}}

其中\alpha是黄金分割的对数：

\alpha=ln(\phi)=ln \frac{1+\sqrt{5}}{2}=0.4812118246

;斐波那契双曲余弦函数
cFh(x)= \frac{2sinh(2x*\alpha)}{\sqrt{5}}

;斐波那契双曲正切函数
tFh(x)=\frac{fsh(x)}{fch(x)}

斐氏双曲函数图
关系式
*sFh(-x)=-sFh(x)

• cFh(-x)=cFh(x-1)
• sFh^2(x)+cFh^2(x)=cFh(2x)
• cFh^2(x)-sFh^2(x)=1+sFh(x)cFh(x)

sFh(x)+sFh(y)=\sqrt(5)sFh(\frac{x+y}{2})cFh(\frac{x-y-1}{2} )

• cFh(x)+cFh(y)=\sqrt{5}cFh(\frac{x+y}{2} )cFh(\frac{x+y-1}{2} )

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级数展开
*fsh(x) \approx \left\{ (4/5\,\sqrt {5}\ln \left( 1/2+1/2\,\sqrt {5} \right) x+{
\frac {8}{15}}\,\sqrt {5} \left( \ln \left( 1/2+1/2\,\sqrt {5}
\right) \right) ^{3}{x}^{3}+{\frac {8}{75}}\,\sqrt {5} \left( \ln
\left( 1/2+1/2\,\sqrt {5} \right) \right) ^{5}{x}^{5}+{\frac {16}{
1575}}\,\sqrt {5} \left( \ln \left( 1/2+1/2\,\sqrt {5} \right)
\right) ^{7}{x}^{7}+O \left( {x}^{9} \right) ) \right\}

fch(x) \approx {(1/5)\sqrt(5)(5+\sqrt(5))/(\sqrt(5)+1)+(2/5)\sqrt(5)ln(1/2+(1/2)\sqrt(5))x+(2/5)\sqrt(5)(5+\sqrt(5))ln(1/2+(1/2)\sqrt(5))^2x^2/(\sqrt(5)+1)+(4/15)\sqrt(5)ln(1/2+(1/2)\sqrt(5))^3x^3+(2/15)\sqrt(5)(5+\sqrt(5))ln(1/2+(1/2)\sqrt(5))^4*x^4/(\sqrt(5)+1)+O(x^5)}

fth(x) \approx {4ln(1/2+(1/2)\sqrt(5))(\sqrt(5)+1)x/(5+\sqrt(5))-8ln(1/2+(1/2)\sqrt(5))^2(\sqrt(5)+1)^2x^2/(5+\sqrt(5))^2+(2(-(8/3)ln(1/2+(1/2)\sqrt(5))^3+8ln(1/2+(1/2)\sqrt(5))^3(\sqrt(5)+1)^2/(5+\sqrt(5))^2))(\sqrt(5)+1)*x^3/(5+\sqrt(5))+O(x^4)}

渐近展开

• sFh(x) \approx \left\{ 1/5\,{\frac {\sqrt {5} \left( \left( \sqrt {5}+1 \right) ^{x

} \right) ^{2}}{ \left( {2}^{x} \right) ^{2}}}-1/5\,{\frac {\sqrt {5}
\left( {2}^{x} \right) ^{2}}{ \left( \left( \sqrt {5}+1 \right) ^{x}
\right) ^{2}}} \right\}

*cFh(x) \approx \left\{ 2/5\,{\frac { \left( 1/4\,\sqrt {5}+1/4 \right) \sqrt {5}
\left( \left( \sqrt {5}+1 \right) ^{x} \right) ^{2}}{ \left( {2}^{x}
\right) ^{2}}}+2/5\,{\frac {\sqrt {5} \left( {2}^{x} \right) ^{2}}{
\left( \sqrt {5}+1 \right) \left( \left( \sqrt {5}+1 \right) ^{x}
\right) ^{2}}} \right\}

Heun函数表示
*sFh(x)=4/5\,\sqrt {5}x \left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0,0,
1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) {\it HeunB} \left( 2,0,0
,0,2\,\sqrt {{\frac {x \left( \sqrt {5}-1 \right) {\it HeunC} \left( 0
,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) }{\sqrt {5}+1}}}
\right) \left( \sqrt {5}+1 \right) ^{-1} \left( {{\rm e}^{2\,{\frac
{x \left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0,0,1/2,{\frac {
\sqrt {5}-1}{\sqrt {5}+1}} \right) }{\sqrt {5}+1}}}} \right) ^{-1}

*cFh(x)=2/5\,i\sqrt {5} \left( 2\,x+1 \right) \left( \sqrt {5}-1 \right) {
\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {{\frac { \left( -2\,x-1
\right) \left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0,0,1/2,{
\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) }{\sqrt {5}+1}}+1/2\,i\pi }
\right) {\it HeunC} \left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}
+1}} \right) \left( \sqrt {5}+1 \right) ^{-1} \left( {{\rm e}^{1/2\,{
\frac {-4\,{\it HeunC} \left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {
5}+1}} \right) x\sqrt {5}+4\,{\it HeunC} \left( 0,1,0,0,1/2,{\frac {
\sqrt {5}-1}{\sqrt {5}+1}} \right) x-2\,{\it HeunC} \left( 0,1,0,0,1/2
,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) \sqrt {5}+2\,{\it HeunC}
\left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) +i\pi
\,\sqrt {5}+i\pi }{\sqrt {5}+1}}}} \right) ^{-1}+1/5\,\sqrt {5}\pi \,{
\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {{\frac { \left( -2\,x-1
\right) \left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0,0,1/2,{
\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) }{\sqrt {5}+1}}+1/2\,i\pi }
\right) \left( {{\rm e}^{1/2\,{\frac {-4\,{\it HeunC} \left( 0,1,0,0
,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) x\sqrt {5}+4\,{\it
HeunC} \left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) x
-2\,{\it HeunC} \left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}}
\right) \sqrt {5}+2\,{\it HeunC} \left( 0,1,0,0,1/2,{\frac {\sqrt {5}
-1}{\sqrt {5}+1}} \right) +i\pi \,\sqrt {5}+i\pi }{\sqrt {5}+1}}}}
\right) ^{-1}

*tFh(x)=4\,x \left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0,0,1/2,{\frac
{\sqrt {5}-1}{\sqrt {5}+1}} \right) {\it HeunB} \left( 2,0,0,0,2\,
\sqrt {{\frac {x \left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0,0
,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) }{\sqrt {5}+1}}}
\right) {{\rm e}^{1/2\,{\frac {-4\,{\it HeunC} \left( 0,1,0,0,1/2,{
\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) x\sqrt {5}+4\,{\it HeunC}
\left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) x-2\,{
\it HeunC} \left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}}
\right) \sqrt {5}+2\,{\it HeunC} \left( 0,1,0,0,1/2,{\frac {\sqrt {5}
-1}{\sqrt {5}+1}} \right) +i\pi \,\sqrt {5}+i\pi }{\sqrt {5}+1}}}}
\left( \sqrt {5}+1 \right) ^{-1} \left( {{\rm e}^{2\,{\frac {x
\left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0,0,1/2,{\frac {
\sqrt {5}-1}{\sqrt {5}+1}} \right) }{\sqrt {5}+1}}}} \right) ^{-1}
\left( {\frac {2\,i \left( 2\,x+1 \right) \left( \sqrt {5}-1
\right) {\it HeunC} \left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}
+1}} \right) }{\sqrt {5}+1}}+\pi \right) ^{-1} \left( {\it HeunB}
\left( 2,0,0,0,\sqrt {2}\sqrt {{\frac { \left( -2\,x-1 \right)
\left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0,0,1/2,{\frac {
\sqrt {5}-1}{\sqrt {5}+1}} \right) }{\sqrt {5}+1}}+1/2\,i\pi }
\right) \right) ^{-1}

斐波那契双曲反函数
;反双曲正弦
*arcsFh(z)=1/2\,{\frac {{\it arcsinh} \left( 1/2\,z\sqrt {5} \right) }{\ln
\left( 1/2+1/2\,\sqrt {5} \right) }}
满足sFh(arcsFh(y))=y

;反双曲余弦

• arccFh(z)= -1/2\,{\frac {\ln \left( 1/2+1/2\,\sqrt {5} \right) -{\it arccosh}

\left( 1/2\,z\sqrt {5} \right) }{\ln \left( 1/2+1/2\,\sqrt {5}
\right) }}
满足arccFh(cFh(z))=z
;反双曲正切
*arctFh(z)= 1/4\,\ln \left( -{\frac {-z+z\sqrt {5}+2}{z+z\sqrt {5}-2}} \right)
\left( \ln \left( 1/2+1/2\,\sqrt {5} \right) \right) ^{-1}
满足arctFh(tFh(z))=z

Heun函数表示
arcsFh(z)=1/2\,z\sqrt {5} \left( \sqrt {5}+1 \right) {\it HeunC} \left( 0,1/2,0,0
,1/4,5\,{\frac {{z}^{2}}{5\,{z}^{2}+4}} \right) {\frac {1}{\sqrt {5\,{
z}^{2}+4}}} \left( \sqrt {5}-1 \right) ^{-1} \left( {\it HeunC}
\left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right)
\right) ^{-1}

arccFh(z)=-1/2+ \left( 1/2\,\sqrt {- \left( -2+z\sqrt {5} \right) ^{2}}z\sqrt {5
} \left( \sqrt {5}+1 \right) {\it HeunC} \left( 0,1/2,0,0,1/4,5/4\,{
\frac {{z}^{2}}{5/4\,{z}^{2}-1}} \right) \left( -2+z\sqrt {5}
\right) ^{-1}{\frac {1}{\sqrt {-5\,{z}^{2}+4}}} \left( \sqrt {5}-1
\right) ^{-1}-1/4\,{\frac {\sqrt {- \left( -2+z\sqrt {5} \right) ^{2}
}\pi \, \left( \sqrt {5}+1 \right) }{ \left( -2+z\sqrt {5} \right)
\left( \sqrt {5}-1 \right) }} \right) \left( {\it HeunC} \left( 0,1,0
,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) \right) ^{-1}

arctFh(x)=z=4\,x \left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0,0,1/2,{
\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) {\it HeunB} \left( 2,0,0,0,2
\,\sqrt {{\frac {x \left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0
,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) }{\sqrt {5}+1}}}
\right) {{\rm e}^{1/2\,{\frac {-4\,{\it HeunC} \left( 0,1,0,0,1/2,{
\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) x\sqrt {5}+4\,{\it HeunC}
\left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) x-2\,{
\it HeunC} \left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}+1}}
\right) \sqrt {5}+2\,{\it HeunC} \left( 0,1,0,0,1/2,{\frac {\sqrt {5}
-1}{\sqrt {5}+1}} \right) +i\pi \,\sqrt {5}+i\pi }{\sqrt {5}+1}}}}
\left( \sqrt {5}+1 \right) ^{-1} \left( {{\rm e}^{2\,{\frac {x
\left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0,0,1/2,{\frac {
\sqrt {5}-1}{\sqrt {5}+1}} \right) }{\sqrt {5}+1}}}} \right) ^{-1}
\left( {\frac {2\,i \left( 2\,x+1 \right) \left( \sqrt {5}-1
\right) {\it HeunC} \left( 0,1,0,0,1/2,{\frac {\sqrt {5}-1}{\sqrt {5}
+1}} \right) }{\sqrt {5}+1}}+\pi \right) ^{-1} \left( {\it HeunB}
\left( 2,0,0,0,\sqrt {2}\sqrt {{\frac { \left( -1-2\,x \right)
\left( \sqrt {5}-1 \right) {\it HeunC} \left( 0,1,0,0,1/2,{\frac {
\sqrt {5}-1}{\sqrt {5}+1}} \right) }{\sqrt {5}+1}}+1/2\,i\pi }
\right) \right) ^{-1}

级数展开
*arcsFh(z) \approx (1/4\,{\frac {\sqrt {5}}{\ln \left( 1/2+1/2\,\sqrt {5} \right) }}z-{
\frac {5}{96}}\,{\frac {\sqrt {5}}{\ln \left( 1/2+1/2\,\sqrt {5}
\right) }}{z}^{3}+{\frac {15}{512}}\,{\frac {\sqrt {5}}{\ln \left( 1
/2+1/2\,\sqrt {5} \right) }}{z}^{5}-{\frac {625}{28672}}\,{\frac {
\sqrt {5}}{\ln \left( 1/2+1/2\,\sqrt {5} \right) }}{z}^{7}+{\frac {
21875}{1179648}}\,{\frac {\sqrt {5}}{\ln \left( 1/2+1/2\,\sqrt {5}
\right) }}{z}^{9}+O \left( {z}^{11} \right) )
>

*arccFh(z) \approx (-1/2\,{\frac {\ln \left( 1/2+1/2\,\sqrt {5} \right) +1/2\,i{\it csgn
} \left( i \left( 1/2\,z\sqrt {5}-1 \right) \right) \pi }{\ln
\left( 1/2+1/2\,\sqrt {5} \right) }}+1/4\,i{\it csgn} \left( i
\left( 1/2\,z\sqrt {5}-1 \right) \right) \sqrt {5} \left( \ln
\left( {\frac {1}{2}}+1/2\,\sqrt {5} \right) \right) ^{-1}z+{\frac {
5}{96}}\,i\sqrt {5}{\it csgn} \left( i \left( 1/2\,z\sqrt {5}-1
\right) \right) \left( \ln \left( {\frac {1}{2}}+1/2\,\sqrt {5}
\right) \right) ^{-1}{z}^{3}+{\frac {15}{512}}\,i\sqrt {5}{\it csgn}
\left( i \left( 1/2\,z\sqrt {5}-1 \right) \right) \left( \ln
\left( {\frac {1}{2}}+1/2\,\sqrt {5} \right) \right) ^{-1}{z}^{5}+{
\frac {625}{28672}}\,i\sqrt {5}{\it csgn} \left( i \left( 1/2\,z\sqrt
{5}-1 \right) \right) \left( \ln \left( {\frac {1}{2}}+1/2\,\sqrt {
5} \right) \right) ^{-1}{z}^{7}+{\frac {21875}{1179648}}\,i\sqrt {5}{
\it csgn} \left( i \left( 1/2\,z\sqrt {5}-1 \right) \right) \left(
\ln \left( {\frac {1}{2}}+1/2\,\sqrt {5} \right) \right) ^{-1}{z}^{9
}+O \left( {z}^{11} \right) )
>

*arctFh(z) \approx (1/4\,{\frac {\sqrt {5}}{\ln \left( 1/2+1/2\,\sqrt {5} \right) }}z+1/
4\,{\frac {1/2\,\sqrt {5} \left( \sqrt {5}+1 \right) -5/2}{\ln
\left( 1/2+1/2\,\sqrt {5} \right) }}{z}^{2}+1/4\,{\frac {-5/6\,\sqrt
{5}-5/2+1/4\,\sqrt {5} \left( \sqrt {5}+1 \right) ^{2}}{\ln \left( 1/
2+1/2\,\sqrt {5} \right) }}{z}^{3}+1/4\,{\frac {-{\frac {5}{16}}\,
\left( \sqrt {5}+1 \right) ^{2}+1/8\,\sqrt {5} \left( \sqrt {5}+1
\right) ^{3}-5/8\,\sqrt {5}-{\frac {25}{8}}}{\ln \left( 1/2+1/2\,
\sqrt {5} \right) }}{z}^{4}+O \left( {z}^{5} \right) )
>

渐近展开
*arcsFh(z) \approx 1/2\,{\frac {1/2\,\ln \left( 5 \right) +\ln \left( z \right) }{\ln
\left( 1/2+1/2\,\sqrt {5} \right) }}+1/10\,{\frac {1}{\ln \left( 1/2
+1/2\,\sqrt {5} \right) {z}^{2}}}-{\frac {3}{100}}\,{\frac {1}{\ln
\left( 1/2+1/2\,\sqrt {5} \right) {z}^{4}}}+{\frac {1}{75}}\,{\frac {
1}{\ln \left( 1/2+1/2\,\sqrt {5} \right) {z}^{6}}}-{\frac {7}{1000}}
\,{\frac {1}{\ln \left( 1/2+1/2\,\sqrt {5} \right) {z}^{8}}}+O
\left( {z}^{-10} \right)

*arccFh(z) \approx -1/2\,{\frac {\ln \left( 1/2+1/2\,\sqrt {5} \right) -1/2\,\ln
\left( 5 \right) -\ln \left( z \right) }{\ln \left( 1/2+1/2\,\sqrt
{5} \right) }}-1/10\,{\frac {1}{\ln \left( 1/2+1/2\,\sqrt {5}
\right) {z}^{2}}}-{\frac {3}{100}}\,{\frac {1}{\ln \left( 1/2+1/2\,
\sqrt {5} \right) {z}^{4}}}-{\frac {1}{75}}\,{\frac {1}{\ln \left( 1/
2+1/2\,\sqrt {5} \right) {z}^{6}}}-{\frac {7}{1000}}\,{\frac {1}{\ln
\left( 1/2+1/2\,\sqrt {5} \right) {z}^{8}}}+O \left( {z}^{-10}
\right)

*arctFh(z) \approx 1/4\,{\frac {\ln \left( {\frac {\sqrt {5}-1}{\sqrt {5}+1}} \right) +i
\pi }{\ln \left( 1/2+1/2\,\sqrt {5} \right) }}+1/4\,{\frac {2+2\,{
\frac {\sqrt {5}-1}{\sqrt {5}+1}}}{ \left( \sqrt {5}-1 \right) \ln
\left( 1/2+1/2\,\sqrt {5} \right) z}}+1/4\,{\frac {8\,{\frac {\sqrt {
5}}{ \left( \sqrt {5}+1 \right) ^{2} \left( \sqrt {5}-1 \right) }}-2\,
{\frac {\sqrt {5} \left( 2+2\,{\frac {\sqrt {5}-1}{\sqrt {5}+1}}
\right) }{ \left( \sqrt {5}+1 \right) \left( \sqrt {5}-1 \right) ^{2
}}}}{\ln \left( 1/2+1/2\,\sqrt {5} \right) {z}^{2}}}+O \left( {z}^{-3
} \right)

反函数图
参考文献