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Hiperbolik funksiyalar

Hiperbolik funksiyalar

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Hiperbolik funksiyalar
Riyazi hesablamalardaHiperbolik funksiyaların törəmələriHiperbolik funksiyaların inteqrallarıLoqarifmaaltı tərs hiperbolik funksiyalarTeylor ardıcıllığı üçün hiperbolik funksiyalarHəmçinin baxXarici keçidlər

Hiperbolik funksiyalar - elementar funksiyalar ailəsindəndir.Triqonometrik funksiyaların analoqu sayılır.Əsas Hiperbolik funksiyalar bunlardır:

  • Hiperbolik sinus
  • Hiperbolik kosinus
  • Hiperbolik tangens
  • Hiperbolik kotangens
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Hiperbolik funksiyalar

Tərs Hiperbolik funksiyalar isə bunlardır:

  • Hiperbolik arksinus
  • Hiperbolik arkskosinus
  • Hiperbolik arkstangens
  • Hiperbolik arkskotangens
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sinh, cosh ve tanh
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csch, sech ve coth

Hiperbolik funksiyalar aşağıdakı funksiyalardan ibarətdir:

  • Hiperbolik sinus:
sinh ⁡ x = e x − e − x 2 = e 2 x − 1 2 e x {\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}} {\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}}
  • Hiperbolik kosinus:
cosh ⁡ x = e x + e − x 2 = e 2 x + 1 2 e x {\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}} {\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}}
  • Hiperbolik tangens:
tanh ⁡ x = sinh ⁡ x cosh ⁡ x = e x − e − x e x + e − x = e 2 x − 1 e 2 x + 1 {\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}} {\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}}
  • Hiperbolik kotangens:
coth ⁡ x = cosh ⁡ x sinh ⁡ x = e x + e − x e x − e − x = e 2 x + 1 e 2 x − 1 {\displaystyle \coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}} {\displaystyle \coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}}
  • Hiperbolik sekans:
sech x = ( cosh ⁡ x ) − 1 = 2 e x + e − x = 2 e x e 2 x + 1 {\displaystyle \operatorname {sech} \,x=\left(\cosh x\right)^{-1}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}} {\displaystyle \operatorname {sech} \,x=\left(\cosh x\right)^{-1}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}}
  • Hiperbolik kosekans:
csch x = ( sinh ⁡ x ) − 1 = 2 e x − e − x = 2 e x e 2 x − 1 {\displaystyle \operatorname {csch} \,x=\left(\sinh x\right)^{-1}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}} {\displaystyle \operatorname {csch} \,x=\left(\sinh x\right)^{-1}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}}

Hiperbolik funksiyalar xəyali vahid (i) dairəsi ilə aşağıdakı kimi də ifade edilir:

  • Hiperbolik sinus:
sinh ⁡ x = − i sin ⁡ i x {\displaystyle \sinh x=-{\rm {i}}\sin {\rm {i}}x\!} {\displaystyle \sinh x=-{\rm {i}}\sin {\rm {i}}x\!}
  • Hiperbolik kosinus:
cosh ⁡ x = cos ⁡ i x {\displaystyle \cosh x=\cos {\rm {i}}x\!} {\displaystyle \cosh x=\cos {\rm {i}}x\!}
  • Hiperbolik tangens:
tanh ⁡ x = − i tan ⁡ i x {\displaystyle \tanh x=-{\rm {i}}\tan {\rm {i}}x\!} {\displaystyle \tanh x=-{\rm {i}}\tan {\rm {i}}x\!}
  • Hiperbolik kotangens:
coth ⁡ x = i cot ⁡ i x {\displaystyle \coth x={\rm {i}}\cot {\rm {i}}x\!} {\displaystyle \coth x={\rm {i}}\cot {\rm {i}}x\!}
  • Hiperbolik sekans:
sech x = sec ⁡ i x {\displaystyle \operatorname {sech} \,x=\sec {{\rm {i}}x}\!} {\displaystyle \operatorname {sech} \,x=\sec {{\rm {i}}x}\!}
  • Hiperbolik kosekans:
csch x = i csc i x {\displaystyle \operatorname {csch} \,x={\rm {i}}\,\csc \,{\rm {i}}x\!} {\displaystyle \operatorname {csch} \,x={\rm {i}}\,\csc \,{\rm {i}}x\!}

i, i2 = −1 - xəyali vahiddir.

d d x sinh ⁡ x = cosh ⁡ x {\displaystyle {\frac {d}{dx}}\sinh x=\cosh x\,} {\displaystyle {\frac {d}{dx}}\sinh x=\cosh x\,}
d d x cosh ⁡ x = sinh ⁡ x {\displaystyle {\frac {d}{dx}}\cosh x=\sinh x\,} {\displaystyle {\frac {d}{dx}}\cosh x=\sinh x\,}
d d x tanh ⁡ x = 1 − tanh 2 ⁡ x = sech 2 x = 1 / cosh 2 ⁡ x {\displaystyle {\frac {d}{dx}}\tanh x=1-\tanh ^{2}x={\hbox{sech}}^{2}x=1/\cosh ^{2}x\,} {\displaystyle {\frac {d}{dx}}\tanh x=1-\tanh ^{2}x={\hbox{sech}}^{2}x=1/\cosh ^{2}x\,}
d d x coth ⁡ x = 1 − coth 2 ⁡ x = − csch 2 x = − 1 / sinh 2 ⁡ x {\displaystyle {\frac {d}{dx}}\coth x=1-\coth ^{2}x=-{\hbox{csch}}^{2}x=-1/\sinh ^{2}x\,} {\displaystyle {\frac {d}{dx}}\coth x=1-\coth ^{2}x=-{\hbox{csch}}^{2}x=-1/\sinh ^{2}x\,}
d d x   csch x = − coth ⁡ x   csch x {\displaystyle {\frac {d}{dx}}\ {\hbox{csch}}\,x=-\coth x\ {\hbox{csch}}\,x\,} {\displaystyle {\frac {d}{dx}}\ {\hbox{csch}}\,x=-\coth x\ {\hbox{csch}}\,x\,}
d d x   sech x = − tanh ⁡ x   sech x {\displaystyle {\frac {d}{dx}}\ {\hbox{sech}}\,x=-\tanh x\ {\hbox{sech}}\,x\,} {\displaystyle {\frac {d}{dx}}\ {\hbox{sech}}\,x=-\tanh x\ {\hbox{sech}}\,x\,}
d d x arsinh x = 1 x 2 + 1 {\displaystyle {\frac {d}{dx}}\,\operatorname {arsinh} \,x={\frac {1}{\sqrt {x^{2}+1}}}} {\displaystyle {\frac {d}{dx}}\,\operatorname {arsinh} \,x={\frac {1}{\sqrt {x^{2}+1}}}}
d d x arcosh x = 1 x 2 − 1 {\displaystyle {\frac {d}{dx}}\,\operatorname {arcosh} \,x={\frac {1}{\sqrt {x^{2}-1}}}} {\displaystyle {\frac {d}{dx}}\,\operatorname {arcosh} \,x={\frac {1}{\sqrt {x^{2}-1}}}}
d d x artanh x = 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\,\operatorname {artanh} \,x={\frac {1}{1-x^{2}}}} {\displaystyle {\frac {d}{dx}}\,\operatorname {artanh} \,x={\frac {1}{1-x^{2}}}}
d d x arcsch x = − 1 | x | 1 + x 2 {\displaystyle {\frac {d}{dx}}\,\operatorname {arcsch} \,x=-{\frac {1}{\left|x\right|{\sqrt {1+x^{2}}}}}} {\displaystyle {\frac {d}{dx}}\,\operatorname {arcsch} \,x=-{\frac {1}{\left|x\right|{\sqrt {1+x^{2}}}}}}
d d x arsech x = − 1 x 1 − x 2 {\displaystyle {\frac {d}{dx}}\,\operatorname {arsech} \,x=-{\frac {1}{x{\sqrt {1-x^{2}}}}}} {\displaystyle {\frac {d}{dx}}\,\operatorname {arsech} \,x=-{\frac {1}{x{\sqrt {1-x^{2}}}}}}
d d x arcoth x = 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\,\operatorname {arcoth} \,x={\frac {1}{1-x^{2}}}} {\displaystyle {\frac {d}{dx}}\,\operatorname {arcoth} \,x={\frac {1}{1-x^{2}}}}
∫ sinh ⁡ a x d x = a − 1 cosh ⁡ a x + C {\displaystyle \int \sinh ax\,dx=a^{-1}\cosh ax+C} {\displaystyle \int \sinh ax\,dx=a^{-1}\cosh ax+C}
∫ cosh ⁡ a x d x = a − 1 sinh ⁡ a x + C {\displaystyle \int \cosh ax\,dx=a^{-1}\sinh ax+C} {\displaystyle \int \cosh ax\,dx=a^{-1}\sinh ax+C}
∫ tanh ⁡ a x d x = a − 1 ln ⁡ ( cosh ⁡ a x ) + C {\displaystyle \int \tanh ax\,dx=a^{-1}\ln(\cosh ax)+C} {\displaystyle \int \tanh ax\,dx=a^{-1}\ln(\cosh ax)+C}
∫ coth ⁡ a x d x = a − 1 ln ⁡ ( sinh ⁡ a x ) + C {\displaystyle \int \coth ax\,dx=a^{-1}\ln(\sinh ax)+C} {\displaystyle \int \coth ax\,dx=a^{-1}\ln(\sinh ax)+C}
∫ d u a 2 + u 2 = sinh − 1 ⁡ ( u a ) + C {\displaystyle \int {\frac {du}{\sqrt {a^{2}+u^{2}}}}=\sinh ^{-1}\left({\frac {u}{a}}\right)+C} {\displaystyle \int {\frac {du}{\sqrt {a^{2}+u^{2}}}}=\sinh ^{-1}\left({\frac {u}{a}}\right)+C}
∫ d u u 2 − a 2 = cosh − 1 ⁡ ( u a ) + C {\displaystyle \int {\frac {du}{\sqrt {u^{2}-a^{2}}}}=\cosh ^{-1}\left({\frac {u}{a}}\right)+C} {\displaystyle \int {\frac {du}{\sqrt {u^{2}-a^{2}}}}=\cosh ^{-1}\left({\frac {u}{a}}\right)+C}
∫ d u a 2 − u 2 = a − 1 tanh − 1 ⁡ ( u a ) + C ; u 2 < a 2 {\displaystyle \int {\frac {du}{a^{2}-u^{2}}}=a^{-1}\tanh ^{-1}\left({\frac {u}{a}}\right)+C;u^{2}<a^{2}} {\displaystyle \int {\frac {du}{a^{2}-u^{2}}}=a^{-1}\tanh ^{-1}\left({\frac {u}{a}}\right)+C;u^{2}<a^{2}}
∫ d u a 2 − u 2 = a − 1 coth − 1 ⁡ ( u a ) + C ; u 2 > a 2 {\displaystyle \int {\frac {du}{a^{2}-u^{2}}}=a^{-1}\coth ^{-1}\left({\frac {u}{a}}\right)+C;u^{2}>a^{2}} {\displaystyle \int {\frac {du}{a^{2}-u^{2}}}=a^{-1}\coth ^{-1}\left({\frac {u}{a}}\right)+C;u^{2}>a^{2}}
∫ d u u a 2 − u 2 = − a − 1 sech − 1 ⁡ ( u a ) + C {\displaystyle \int {\frac {du}{u{\sqrt {a^{2}-u^{2}}}}}=-a^{-1}\operatorname {sech} ^{-1}\left({\frac {u}{a}}\right)+C} {\displaystyle \int {\frac {du}{u{\sqrt {a^{2}-u^{2}}}}}=-a^{-1}\operatorname {sech} ^{-1}\left({\frac {u}{a}}\right)+C}
∫ d u u a 2 + u 2 = − a − 1 csch − 1 ⁡ | u a | + C {\displaystyle \int {\frac {du}{u{\sqrt {a^{2}+u^{2}}}}}=-a^{-1}\operatorname {csch} ^{-1}\left|{\frac {u}{a}}\right|+C} {\displaystyle \int {\frac {du}{u{\sqrt {a^{2}+u^{2}}}}}=-a^{-1}\operatorname {csch} ^{-1}\left|{\frac {u}{a}}\right|+C}

C sabit ədəddir.

arsinh x = ln ⁡ ( x + x 2 + 1 ) {\displaystyle \operatorname {arsinh} \,x=\ln \left(x+{\sqrt {x^{2}+1}}\right)} {\displaystyle \operatorname {arsinh} \,x=\ln \left(x+{\sqrt {x^{2}+1}}\right)}
arcosh x = ln ⁡ ( x + x 2 − 1 ) ; x ≥ 1 {\displaystyle \operatorname {arcosh} \,x=\ln \left(x+{\sqrt {x^{2}-1}}\right);x\geq 1} {\displaystyle \operatorname {arcosh} \,x=\ln \left(x+{\sqrt {x^{2}-1}}\right);x\geq 1}
artanh x = 1 2 ln ⁡ 1 + x 1 − x ; | x | < 1 {\displaystyle \operatorname {artanh} \,x={\tfrac {1}{2}}\ln {\frac {1+x}{1-x}};\left|x\right|<1} {\displaystyle \operatorname {artanh} \,x={\tfrac {1}{2}}\ln {\frac {1+x}{1-x}};\left|x\right|<1}
arcoth x = 1 2 ln ⁡ x + 1 x − 1 ; | x | > 1 {\displaystyle \operatorname {arcoth} \,x={\tfrac {1}{2}}\ln {\frac {x+1}{x-1}};\left|x\right|>1} {\displaystyle \operatorname {arcoth} \,x={\tfrac {1}{2}}\ln {\frac {x+1}{x-1}};\left|x\right|>1}
arsech x = ln ⁡ 1 + 1 − x 2 x ; 0 < x ≤ 1 {\displaystyle \operatorname {arsech} \,x=\ln {\frac {1+{\sqrt {1-x^{2}}}}{x}};0<x\leq 1} {\displaystyle \operatorname {arsech} \,x=\ln {\frac {1+{\sqrt {1-x^{2}}}}{x}};0<x\leq 1}
arcsch x = ln ⁡ ( 1 x + 1 + x 2 | x | ) {\displaystyle \operatorname {arcsch} \,x=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1+x^{2}}}{\left|x\right|}}\right)} {\displaystyle \operatorname {arcsch} \,x=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1+x^{2}}}{\left|x\right|}}\right)}
sinh ⁡ x = x + x 3 3 ! + x 5 5 ! + x 7 7 ! + ⋯ = ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 1 ) ! {\displaystyle \sinh x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}} {\displaystyle \sinh x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}}
cosh ⁡ x = 1 + x 2 2 ! + x 4 4 ! + x 6 6 ! + ⋯ = ∑ n = 0 ∞ x 2 n ( 2 n ) ! {\displaystyle \cosh x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}} {\displaystyle \cosh x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}}
tanh ⁡ x = x − x 3 3 + 2 x 5 15 − 17 x 7 315 + ⋯ = ∑ n = 1 ∞ 2 2 n ( 2 2 n − 1 ) B 2 n x 2 n − 1 ( 2 n ) ! , | x | < π 2 {\displaystyle \tanh x=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}} {\displaystyle \tanh x=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}}
coth ⁡ x = x − 1 + x 3 − x 3 45 + 2 x 5 945 + ⋯ = x − 1 + ∑ n = 1 ∞ 2 2 n B 2 n x 2 n − 1 ( 2 n ) ! , 0 < | x | < π {\displaystyle \coth x=x^{-1}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\cdots =x^{-1}+\sum _{n=1}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi } {\displaystyle \coth x=x^{-1}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\cdots =x^{-1}+\sum _{n=1}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi } (Laurent ardıcıllığı)
sech x = 1 − x 2 2 + 5 x 4 24 − 61 x 6 720 + ⋯ = ∑ n = 0 ∞ E 2 n x 2 n ( 2 n ) ! , | x | < π 2 {\displaystyle \operatorname {sech} \,x=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\cdots =\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}} {\displaystyle \operatorname {sech} \,x=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\cdots =\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}}
csch x = x − 1 − x 6 + 7 x 3 360 − 31 x 5 15120 + ⋯ = x − 1 + ∑ n = 1 ∞ 2 ( 1 − 2 2 n − 1 ) B 2 n x 2 n − 1 ( 2 n ) ! , 0 < | x | < π {\displaystyle \operatorname {csch} \,x=x^{-1}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\cdots =x^{-1}+\sum _{n=1}^{\infty }{\frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi } {\displaystyle \operatorname {csch} \,x=x^{-1}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\cdots =x^{-1}+\sum _{n=1}^{\infty }{\frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi } (Laurent ardıcıllığı)
B n {\displaystyle B_{n}\,} {\displaystyle B_{n}\,} ninci Bernoulli sayıdır.
E n {\displaystyle E_{n}\,} {\displaystyle E_{n}\,} ninci Eyler sayıdır.
  • Hiperbola
Vikianbarda Hiperbolik funksiyalar ilə əlaqəli mediafayllar var.
  • Hiperbolik fonksiyonlar Arxivləşdirilib 2012-02-18 at the Wayback Machine PlanetMath
  • Hiperbolik fonksiyonlar (MathWorld)
  • GonioLab Arxivləşdirilib 2007-10-06 at the Wayback Machine: Birim çember, trigonometrik ve hiperbolik fonksiyonların gösterimi (Java Web Start)
  • Web-tabanlı hiperbolik fonksiyon hesap makinesi
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Riyazi hesablamalarda

Hiperbolik funksiyaların törəmələri

Hiperbolik funksiyaların inteqralları

Loqarifmaaltı tərs hiperbolik funksiyalar

Teylor ardıcıllığı üçün hiperbolik funksiyalar

Həmçinin bax

Xarici keçidlər