Hiperbolik funksiyalar elementar funksiyalar ailəsindəndir Triqonometrik funksiyaların analoqu sayılır əsas Hiperbolik f

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

Riyazi hesablamalarda

sinh, cosh ve tanh

csch, sech ve coth

Hiperbolik funksiyalar aşağıdakı funksiyalardan ibarətdir:

Hiperbolik sinus:

\sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}

Hiperbolik kosinus:

\cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}

Hiperbolik tangens:

\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={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}

Hiperbolik sekans:

\operatorname {sech} \,x=\left(\cosh x\right)^{-1}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}

Hiperbolik kosekans:

\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=-{\rm {i}}\sin {\rm {i}}x\!

Hiperbolik kosinus:

\cosh x=\cos {\rm {i}}x\!

Hiperbolik tangens:

\tanh x=-{\rm {i}}\tan {\rm {i}}x\!

Hiperbolik kotangens:

\coth x={\rm {i}}\cot {\rm {i}}x\!

Hiperbolik sekans:

\operatorname {sech} \,x=\sec {{\rm {i}}x}\!

Hiperbolik kosekans:

\operatorname {csch} \,x={\rm {i}}\,\csc \,{\rm {i}}x\!

i, i² = −1 - xəyali vahiddir.

Hiperbolik funksiyaların törəmələri

{\frac {d}{dx}}\sinh x=\cosh x\,

{\frac {d}{dx}}\cosh x=\sinh x\,

{\frac {d}{dx}}\tanh x=1-\tanh ^{2}x={\hbox{sech}}^{2}x=1/\cosh ^{2}x\,

{\frac {d}{dx}}\coth x=1-\coth ^{2}x=-{\hbox{csch}}^{2}x=-1/\sinh ^{2}x\,

{\frac {d}{dx}}\ {\hbox{csch}}\,x=-\coth x\ {\hbox{csch}}\,x\,

{\frac {d}{dx}}\ {\hbox{sech}}\,x=-\tanh x\ {\hbox{sech}}\,x\,

{\frac {d}{dx}}\,\operatorname {arsinh} \,x={\frac {1}{\sqrt {x^{2}+1}}}

{\frac {d}{dx}}\,\operatorname {arcosh} \,x={\frac {1}{\sqrt {x^{2}-1}}}

{\frac {d}{dx}}\,\operatorname {artanh} \,x={\frac {1}{1-x^{2}}}

{\frac {d}{dx}}\,\operatorname {arcsch} \,x=-{\frac {1}{\left|x\right|{\sqrt {1+x^{2}}}}}

{\frac {d}{dx}}\,\operatorname {arsech} \,x=-{\frac {1}{x{\sqrt {1-x^{2}}}}}

{\frac {d}{dx}}\,\operatorname {arcoth} \,x={\frac {1}{1-x^{2}}}

Hiperbolik funksiyaların inteqralları

\int \sinh ax\,dx=a^{-1}\cosh ax+C

\int \cosh ax\,dx=a^{-1}\sinh ax+C

\int \tanh ax\,dx=a^{-1}\ln(\cosh ax)+C

\int \coth ax\,dx=a^{-1}\ln(\sinh ax)+C

\int {\frac {du}{\sqrt {a^{2}+u^{2}}}}=\sinh ^{-1}\left({\frac {u}{a}}\right)+C

\int {\frac {du}{\sqrt {u^{2}-a^{2}}}}=\cosh ^{-1}\left({\frac {u}{a}}\right)+C

\int {\frac {du}{a^{2}-u^{2}}}=a^{-1}\tanh ^{-1}\left({\frac {u}{a}}\right)+C;u^{2}<a^{2}

\int {\frac {du}{a^{2}-u^{2}}}=a^{-1}\coth ^{-1}\left({\frac {u}{a}}\right)+C;u^{2}>a^{2}

\int {\frac {du}{u{\sqrt {a^{2}-u^{2}}}}}=-a^{-1}\operatorname {sech} ^{-1}\left({\frac {u}{a}}\right)+C

\int {\frac {du}{u{\sqrt {a^{2}+u^{2}}}}}=-a^{-1}\operatorname {csch} ^{-1}\left|{\frac {u}{a}}\right|+C

C sabit ədəddir.

Loqarifmaaltı tərs hiperbolik funksiyalar

\operatorname {arsinh} \,x=\ln \left(x+{\sqrt {x^{2}+1}}\right)

\operatorname {arcosh} \,x=\ln \left(x+{\sqrt {x^{2}-1}}\right);x\geq 1

\operatorname {artanh} \,x={\tfrac {1}{2}}\ln {\frac {1+x}{1-x}};\left|x\right|<1

\operatorname {arcoth} \,x={\tfrac {1}{2}}\ln {\frac {x+1}{x-1}};\left|x\right|>1

\operatorname {arsech} \,x=\ln {\frac {1+{\sqrt {1-x^{2}}}}{x}};0<x\leq 1

\operatorname {arcsch} \,x=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1+x^{2}}}{\left|x\right|}}\right)

Teylor ardıcıllığı üçün hiperbolik funksiyalar

\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+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}

\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}+{\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)

\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}}

\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}\,

ninci Bernoulli sayıdır.

E_{n}\,

ninci Eyler sayıdır.

Həmçinin bax

Hiperbola

Xarici keçidlər

Vikianbarda Hiperbolik funksiyalar ilə əlaqəli mediafayllar var.

Hiperbolik fonksiyonlar 2012-02-18 at the Wayback Machine PlanetMath
Hiperbolik fonksiyonlar (MathWorld)
GonioLab 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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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 arkskotangensRiyazi hesablamalardasinh cosh ve tanhcsch sech ve coth Hiperbolik funksiyalar asagidaki funksiyalardan ibaretdir Hiperbolik sinus sinh x ex e x2 e2x 12ex displaystyle sinh x frac e x e x 2 frac e 2x 1 2e x dd Hiperbolik kosinus cosh x ex e x2 e2x 12ex displaystyle cosh x frac e x e x 2 frac e 2x 1 2e x dd Hiperbolik tangens tanh x sinh xcosh x ex e xex e x e2x 1e2x 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 xsinh x ex e xex e x e2x 1e2x 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 sechx cosh x 1 2ex e x 2exe2x 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 cschx sinh x 1 2ex e x 2exe2x 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 isin ix displaystyle sinh x rm i sin rm i x dd Hiperbolik kosinus cosh x cos ix displaystyle cosh x cos rm i x dd Hiperbolik tangens tanh x itan ix displaystyle tanh x rm i tan rm i x dd Hiperbolik kotangens coth x icot ix displaystyle coth x rm i cot rm i x dd Hiperbolik sekans sechx sec ix displaystyle operatorname sech x sec rm i x dd Hiperbolik kosekans cschx icscix displaystyle operatorname csch x rm i csc rm i x dd i i2 1 xeyali vahiddir Hiperbolik funksiyalarin toremeleriddxsinh x cosh x displaystyle frac d dx sinh x cosh x ddxcosh x sinh x displaystyle frac d dx cosh x sinh x ddxtanh x 1 tanh2 x sech2x 1 cosh2 x displaystyle frac d dx tanh x 1 tanh 2 x hbox sech 2 x 1 cosh 2 x ddxcoth x 1 coth2 x csch2x 1 sinh2 x displaystyle frac d dx coth x 1 coth 2 x hbox csch 2 x 1 sinh 2 x ddx cschx coth x cschx displaystyle frac d dx hbox csch x coth x hbox csch x ddx sechx tanh x sechx displaystyle frac d dx hbox sech x tanh x hbox sech x ddxarsinhx 1x2 1 displaystyle frac d dx operatorname arsinh x frac 1 sqrt x 2 1 ddxarcoshx 1x2 1 displaystyle frac d dx operatorname arcosh x frac 1 sqrt x 2 1 ddxartanhx 11 x2 displaystyle frac d dx operatorname artanh x frac 1 1 x 2 ddxarcschx 1 x 1 x2 displaystyle frac d dx operatorname arcsch x frac 1 left x right sqrt 1 x 2 ddxarsechx 1x1 x2 displaystyle frac d dx operatorname arsech x frac 1 x sqrt 1 x 2 ddxarcothx 11 x2 displaystyle frac d dx operatorname arcoth x frac 1 1 x 2 Hiperbolik funksiyalarin inteqrallari sinh axdx a 1cosh ax C displaystyle int sinh ax dx a 1 cosh ax C cosh axdx a 1sinh ax C displaystyle int cosh ax dx a 1 sinh ax C tanh axdx a 1ln cosh ax C displaystyle int tanh ax dx a 1 ln cosh ax C coth axdx a 1ln sinh ax C displaystyle int coth ax dx a 1 ln sinh ax C dua2 u2 sinh 1 ua C displaystyle int frac du sqrt a 2 u 2 sinh 1 left frac u a right C duu2 a2 cosh 1 ua C displaystyle int frac du sqrt u 2 a 2 cosh 1 left frac u a right C dua2 u2 a 1tanh 1 ua C u2 lt a2 displaystyle int frac du a 2 u 2 a 1 tanh 1 left frac u a right C u 2 lt a 2 dua2 u2 a 1coth 1 ua C u2 gt a2 displaystyle int frac du a 2 u 2 a 1 coth 1 left frac u a right C u 2 gt a 2 duua2 u2 a 1sech 1 ua C displaystyle int frac du u sqrt a 2 u 2 a 1 operatorname sech 1 left frac u a right C duua2 u2 a 1csch 1 ua 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 funksiyalararsinhx ln x x2 1 displaystyle operatorname arsinh x ln left x sqrt x 2 1 right arcoshx ln x x2 1 x 1 displaystyle operatorname arcosh x ln left x sqrt x 2 1 right x geq 1 artanhx 12ln 1 x1 x x lt 1 displaystyle operatorname artanh x tfrac 1 2 ln frac 1 x 1 x left x right lt 1 arcothx 12ln x 1x 1 x gt 1 displaystyle operatorname arcoth x tfrac 1 2 ln frac x 1 x 1 left x right gt 1 arsechx ln 1 1 x2x 0 lt x 1 displaystyle operatorname arsech x ln frac 1 sqrt 1 x 2 x 0 lt x leq 1 arcschx ln 1x 1 x2 x displaystyle operatorname arcsch x ln left frac 1 x frac sqrt 1 x 2 left x right right Teylor ardicilligi ucun hiperbolik funksiyalarsinh x x x33 x55 x77 n 0 x2n 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 x22 x44 x66 n 0 x2n 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 x33 2x515 17x7315 n 1 22n 22n 1 B2nx2n 1 2n x lt p2 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 x3 x345 2x5945 x 1 n 1 22nB2nx2n 1 2n 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 sechx 1 x22 5x424 61x6720 n 0 E2nx2n 2n x lt p2 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 cschx x 1 x6 7x3360 31x515120 x 1 n 1 2 1 22n 1 B2nx2n 1 2n 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 Bn displaystyle B n ninci Bernoulli sayidir En displaystyle E n ninci Eyler sayidir Hemcinin baxHiperbolaXarici kecidlerVikianbarda Hiperbolik funksiyalar ile elaqeli mediafayllar var Hiperbolik fonksiyonlar 2012 02 18 at the Wayback Machine PlanetMath Hiperbolik fonksiyonlar MathWorld GonioLab 2007 10 06 at the Wayback Machine Birim cember trigonometrik ve hiperbolik fonksiyonlarin gosterimi Java Web Start Web tabanli hiperbolik fonksiyon hesap makinesi

Nəşr tarixi: İyun 15, 2024, 21:31 pm