fbpx
Wikipedia

Hiperbolik funksiyalar

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

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

Tərs Hiperbolik funksiyalar isə bunlardır:

  • Hiperbolik arksinus
  • Hiperbolik arkskosinus
  • Hiperbolik arkstangens
  • Hiperbolik arkskotangens

Mündəricat

sinh, cosh ve tanh
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}}}}
  • 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}}}}
  • 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}}}
  • 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}}}
  • 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}}}
  • 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}}}

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\!}
  • Hiperbolik kosinus:
cosh x = cos 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\!}
  • Hiperbolik kotangens:
coth x = i cot 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}\!}
  • Hiperbolik kosekans:
csch x = i csc 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\,}
d d x 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\,}
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\,}
d d x csch x = coth x 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\,}
d d x arsinh x = 1 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}}}}
d d x artanh x = 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}}}}}}
d d x arsech x = 1 x 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}}}}
sinh a x d x = a 1 cosh a x + 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}
tanh a x d x = a 1 ln ( cosh a x ) + 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}
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}
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}
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}}
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}}
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}
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}

C sabit ədəddir.

arsinh x = ln ( x + x 2 + 1 ) {\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}
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}
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}
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}
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)}
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)!}}}
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)!}}}
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}}}
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 } (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}}}
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 } (Laurent ardıcıllığı)
B n {\displaystyle B_{n}\,} ninci Bernoulli sayıdır.
E n {\displaystyle E_{n}\,} ninci Eyler sayıdır.
Vikianbarda Hiperbolik funksiyalar ilə əlaqəli mediafayllar var.
  • PlanetMath
  • (MathWorld)
  • 2007-10-06 at the Wayback Machine: Birim çember, trigonometrik ve hiperbolik fonksiyonların gösterimi (Java Web Start)

Hiperbolik funksiyalar
hiperbolik, funksiyalar, elementar, funksiyalar, ailəsindəndir, triqonometrik, funksiyaların, analoqu, sayılır, əsas, bunlardır, hiperbolik, sinus, hiperbolik, kosinus, hiperbolik, tangens, hiperbolik, kotangens, tərs, isə, bunlardır, hiperbolik, arksinus, hip. Hiperbolik funksiyalar Dil Izle Redakte Hiperbolik funksiyalar elementar funksiyalar ailesindendir Triqonometrik funksiyalarin analoqu sayilir Esas Hiperbolik funksiyalar bunlardir Hiperbolik sinus Hiperbolik kosinus Hiperbolik tangens Hiperbolik kotangensHiperbolik funksiyalar Ters Hiperbolik funksiyalar ise bunlardir Hiperbolik arksinus Hiperbolik arkskosinus Hiperbolik arkstangens Hiperbolik arkskotangensMundericat 1 Riyazi hesablamalarda 2 Hiperbolik funksiyalarin toremeleri 3 Hiperbolik funksiyalarin inteqrallari 4 Loqarifmaalti ters hiperbolik funksiyalar 5 Teylor ardicilligi ucun hiperbolik funksiyalar 6 Hemcinin bax 7 Xarici kecidlerRiyazi hesablamalarda Redakte sinh cosh ve tanh csch sech ve coth Hiperbolik funksiyalar asagidaki funksiyalardan ibaretdir 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 dd 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 dd 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 dd 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 dd 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 dd 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 dd Hiperbolik funksiyalar xeyali vahid i dairesi ile asagidaki kimi de ifade edilir Hiperbolik sinus sinh x i sin i x displaystyle sinh x rm i sin rm i x dd Hiperbolik kosinus cosh x cos i x displaystyle cosh x cos rm i x dd Hiperbolik tangens tanh x i tan i x displaystyle tanh x rm i tan rm i x dd Hiperbolik kotangens coth x i cot i x displaystyle coth x rm i cot rm i x dd Hiperbolik sekans sech x sec i x displaystyle operatorname sech x sec rm i x dd Hiperbolik kosekans csch x i csc i x displaystyle operatorname csch x rm i csc rm i x dd i i2 1 xeyali vahiddir Hiperbolik funksiyalarin toremeleri Redakted d x 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 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 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 d d x csch x coth x 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 d d x arsinh x 1 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 d d x artanh x 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 d d x arsech x 1 x 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 Hiperbolik funksiyalarin inteqrallari Redakte sinh a x d x a 1 cosh a x 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 tanh a x d x a 1 ln cosh a x 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 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 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 d u a 2 u 2 a 1 tanh 1 u a C u 2 lt a 2 displaystyle int frac du a 2 u 2 a 1 tanh 1 left frac u a right C u 2 lt a 2 d u a 2 u 2 a 1 coth 1 u a C u 2 gt a 2 displaystyle int frac du a 2 u 2 a 1 coth 1 left frac u a right C u 2 gt 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 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 C sabit ededdir Loqarifmaalti ters hiperbolik funksiyalar Redaktearsinh x ln x x 2 1 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 artanh x 1 2 ln 1 x 1 x x lt 1 displaystyle operatorname artanh x tfrac 1 2 ln frac 1 x 1 x left x right lt 1 arcoth x 1 2 ln x 1 x 1 x gt 1 displaystyle operatorname arcoth x tfrac 1 2 ln frac x 1 x 1 left x right gt 1 arsech x ln 1 1 x 2 x 0 lt x 1 displaystyle operatorname arsech x ln frac 1 sqrt 1 x 2 x 0 lt 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 Teylor ardicilligi ucun hiperbolik funksiyalar Redaktesinh 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 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 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 lt p 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 lt 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 lt x lt p 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 lt left x right lt pi Laurent ardicilligi 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 lt p 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 lt 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 lt x lt p 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 lt left x right lt pi Laurent ardicilligi B n displaystyle B n ninci Bernoulli sayidir E n displaystyle E n ninci Eyler sayidir Hemcinin bax RedakteHiperbolaXarici kecidler Redakte Vikianbarda Hiperbolik funksiyalar ile elaqeli mediafayllar var Hiperbolik fonksiyonlar PlanetMath Hiperbolik fonksiyonlar MathWorld GonioLab Arxivlesdirilib 2007 10 06 at the Wayback Machine Birim cember trigonometrik ve hiperbolik fonksiyonlarin gosterimi Java Web Start Web tabanli hiperbolik fonksiyon hesap makinesiMenbe https az wikipedia org w index php title Hiperbolik funksiyalar amp oldid 5730046, wikipedia, oxu, kitab, kitabxana, axtar, tap, hersey,

ne axtarsan burda

, en yaxsi meqale sayti, meqaleler, kitablar, oyrenmek, wiki, bilgi, tarix, seks, porno, indir, yukle, sex, azeri sex, azeri, seks yukle, sex yukle, izle, seks izle, porno izle, mobil seks, telefon ucun, chat, azeri chat, tanisliq, tanishliq, azeri tanishliq, sayt, medeni, medeni saytlar, chatlar, mekan, tanisliq mekani, mekanlari, yüklə, pulsuz, pulsuz yüklə, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, şəkil, muisiqi, mahnı, kino, film, kitab, oyun, oyunlar.