التدوين أكثر شيوعا وتلك المحددة من قبل ISO 80000-2 هو تسمية الدوال الزائدية العكسية باستخدام البادئة ar- area التي تعني "مساحة") لأن عمدتها هي عبارة عن مساحة القطاع الزائدي المحدد بنصفي مستقيم ، مثال: ar sinh ،ar cosh.
يفضل مؤلفون آخرون استخدام التدوين (arg sinh، وarg cosh، وarg tanh)، حيث البادئة arg- argumentum [ 10] عمدة الجيب الزائدي ، عمدة جيب تمام الزائدي ، ... وهكذا.
في علوم الحاسوب ، تُختصَر الدوال غالبًا باستخدام البادئة a- a sinh.
عدل
دالة معرفة على جميع الأعداد الحقيقية بـ:
arsinh
x
=
ln
(
x
+
x
2
+
1
)
{\displaystyle \operatorname {arsinh} x=\ln \left(x+{\sqrt {x^{2}+1}}\right)}
دالة معرفة على المجال :
[
1
,
+
∞
[
{\displaystyle [1,+\infty [}
arcosh
x
=
ln
(
x
+
x
2
−
1
)
{\displaystyle \operatorname {arcosh} x=\ln \left(x+{\sqrt {x^{2}-1}}\right)}
دالة معرفة على المجال
]
−
1
,
1
[
{\displaystyle ]-1,1[}
artanh
x
=
1
2
ln
(
1
+
x
1
−
x
)
{\displaystyle \operatorname {artanh} x={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)}
عدل
دالة معرفة على المجال
]
−
∞
,
−
1
[
∪
]
1
,
+
∞
[
{\displaystyle ]-\infty ,-1[\cup ]1,+\infty [}
arcoth
x
=
1
2
ln
(
x
+
1
x
−
1
)
{\displaystyle \operatorname {arcoth} x={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)}
دالة معرفة على المجال
]
0
,
1
]
{\displaystyle ]0,1]}
arsech
x
=
ln
(
1
x
+
1
x
2
−
1
)
=
ln
(
1
+
1
−
x
2
x
)
{\displaystyle \operatorname {arsech} x=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}-1}}\right)=\ln \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)}
عدل
دالة معرفة على جميع الأعداد الحقيقية ما عدا الصفر بـ:
arcsch
x
=
ln
(
1
x
+
1
x
2
+
1
)
=
ln
(
1
x
+
1
+
x
2
|
x
|
)
{\displaystyle \operatorname {arcsch} x=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}+1}}\right)=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1+x^{2}}}{\left|x\right|}}\right)}
∀
x
∈
[
1
,
+
∞
[
:
arcosh
(
x
)
=
ln
(
x
+
x
2
−
1
)
{\displaystyle \forall x\in [1,+\infty [\;:\;\operatorname {arcosh} (x)=\ln(x+{\sqrt {x^{2}-1}})}
نضع:
y
=
arcosh
(
x
)
;
x
≥
1
x
=
cosh
(
y
)
;
y
≥
0
{\displaystyle {\begin{aligned}y&=\operatorname {arcosh} (x);\quad x\geq 1\\x&=\operatorname {cosh} (y);\quad \ \ \ \ y\geq 0\end{aligned}}}
لدينا:
cosh
(
y
)
+
sinh
(
y
)
=
e
y
.
.
.
.
(
∗
)
{\displaystyle \cosh(y)+\sinh(y)=e^{y}....(*)}
و
cosh
2
(
y
)
−
sinh
2
(
y
)
=
1
{\displaystyle \cosh ^{2}(y)-\sinh ^{2}(y)=1}
إذن :
sinh
2
(
y
)
=
cosh
2
(
y
)
−
1
{\displaystyle \sinh ^{2}(y)=\cosh ^{2}(y)-1}
ومنه:
|
sinh
(
y
)
|
=
cosh
2
(
y
)
−
1
;
y
≥
0
⇒
sinh
(
y
)
≥
0
sinh
(
y
)
=
cosh
2
(
y
)
−
1
(
∗
)
:
cosh
(
y
)
+
cosh
2
(
y
)
−
1
=
e
y
⇒
y
=
ln
(
cosh
(
y
)
+
cosh
2
(
y
)
−
1
)
arcosh
(
x
)
=
ln
(
x
+
x
2
−
1
)
{\displaystyle {\begin{aligned}|\sinh(y)|&={\sqrt {\cosh ^{2}(y)-1}}\;;\;y\geq 0\Rightarrow \sinh(y)\geq 0\\\sinh(y)&={\sqrt {\cosh ^{2}(y)-1}}\\(*)&:\cosh(y)+{\sqrt {\cosh ^{2}(y)-1}}=e^{y}\\\Rightarrow y&=\ln \left(\cosh(y)+{\sqrt {\cosh ^{2}(y)-1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right)\end{aligned}}}
نعتبر دالة جيب التمام العكسية التالية :
y
=
arcosh
x
{\displaystyle y=\operatorname {arcosh} x}
بالتعريف:
x
=
cosh
y
=
e
y
+
e
−
y
2
{\displaystyle x=\cosh y={\frac {e^{y}+e^{-y}}{2}}}
2
x
=
e
y
+
e
−
y
{\displaystyle 2x={e^{y}+e^{-y}}}
e
y
−
2
x
+
e
−
y
=
0
{\displaystyle e^{y}-2x+e^{-y}=0}
e
y
−
2
x
+
1
e
y
=
0
{\displaystyle e^{y}-2x+{\frac {1}{e^{y}}}=0}
e
2
y
−
2
x
e
y
+
1
=
0
{\displaystyle e^{2y}-2xe^{y}+1=0}
نضع
u
=
e
y
{\displaystyle u=e^{y}}
u
2
−
2
x
u
+
1
=
0
{\displaystyle u^{2}-2xu+1=0}
نحل المعادلة من الدرجة الثانية:
u
=
2
x
±
4
x
2
−
4
2
{\displaystyle u={\frac {2x\pm {\sqrt {4x^{2}-4}}}{2}}}
e
y
=
2
x
±
4
x
2
−
4
2
{\displaystyle e^{y}={\frac {2x\pm {\sqrt {4x^{2}-4}}}{2}}}
e
y
=
2
x
±
4
(
x
2
−
1
)
2
{\displaystyle e^{y}={\frac {2x\pm {\sqrt {4(x^{2}-1)}}}{2}}}
2
e
y
=
2
x
±
2
x
2
−
1
{\displaystyle 2e^{y}={2x\pm 2{\sqrt {x^{2}-1}}}}
e
y
=
x
±
x
2
−
1
{\displaystyle e^{y}={x\pm {\sqrt {x^{2}-1}}}}
ندخل اللوغاريتم الطبيعي الطرفين:
ln
e
y
=
ln
(
x
±
x
2
−
1
)
{\displaystyle \ln e^{y}=\ln \left(x\pm {\sqrt {x^{2}-1}}\right)}
y
=
ln
(
x
+
x
2
−
1
)
{\displaystyle y=\ln \left(x+{\sqrt {x^{2}-1}}\right)}
ومنه نستنتج أن:
arcosh
x
=
ln
(
x
+
x
2
−
1
)
{\displaystyle \operatorname {arcosh} x=\ln \left(x+{\sqrt {x^{2}-1}}\right)}
arsinh
u
±
arsinh
v
=
arsinh
(
u
1
+
v
2
±
v
1
+
u
2
)
{\displaystyle \operatorname {arsinh} u\pm \operatorname {arsinh} v=\operatorname {arsinh} \left(u{\sqrt {1+v^{2}}}\pm v{\sqrt {1+u^{2}}}\right)}
arcosh
u
±
arcosh
v
=
arcosh
(
u
v
±
(
u
2
−
1
)
(
v
2
−
1
)
)
{\displaystyle \operatorname {arcosh} u\pm \operatorname {arcosh} v=\operatorname {arcosh} \left(uv\pm {\sqrt {(u^{2}-1)(v^{2}-1)}}\right)}
artanh
u
±
artanh
v
=
artanh
(
u
±
v
1
±
u
v
)
{\displaystyle \operatorname {artanh} u\pm \operatorname {artanh} v=\operatorname {artanh} \left({\frac {u\pm v}{1\pm uv}}\right)}
arsinh
u
+
arcosh
v
=
arsinh
(
u
v
+
(
1
+
u
2
)
(
v
2
−
1
)
)
=
arcosh
(
v
1
+
u
2
+
u
v
2
−
1
)
{\displaystyle {\begin{aligned}\operatorname {arsinh} u+\operatorname {arcosh} v&=\operatorname {arsinh} \left(uv+{\sqrt {(1+u^{2})(v^{2}-1)}}\right)\\&=\operatorname {arcosh} \left(v{\sqrt {1+u^{2}}}+u{\sqrt {v^{2}-1}}\right)\end{aligned}}}
عدل
sinh
(
arcosh
x
)
=
x
2
−
1
for
|
x
|
>
1
sinh
(
artanh
x
)
=
x
1
−
x
2
for
−
1
<
x
<
1
cosh
(
arsinh
x
)
=
1
+
x
2
cosh
(
artanh
x
)
=
1
1
−
x
2
for
−
1
<
x
<
1
tanh
(
arsinh
x
)
=
x
1
+
x
2
tanh
(
arcosh
x
)
=
x
2
−
1
x
for
|
x
|
>
1
{\displaystyle {\begin{aligned}&\sinh(\operatorname {arcosh} x)={\sqrt {x^{2}-1}}\quad {\text{for}}\quad |x|>1\\&\sinh(\operatorname {artanh} x)={\frac {x}{\sqrt {1-x^{2}}}}\quad {\text{for}}\quad -1<x<1\\&\cosh(\operatorname {arsinh} x)={\sqrt {1+x^{2}}}\\&\cosh(\operatorname {artanh} x)={\frac {1}{\sqrt {1-x^{2}}}}\quad {\text{for}}\quad -1<x<1\\&\tanh(\operatorname {arsinh} x)={\frac {x}{\sqrt {1+x^{2}}}}\\&\tanh(\operatorname {arcosh} x)={\frac {\sqrt {x^{2}-1}}{x}}\quad {\text{for}}\quad |x|>1\end{aligned}}}
d
d
x
arsinh
x
=
1
x
2
+
1
,
for all real
x
d
d
x
arcosh
x
=
1
x
2
−
1
,
for all real
x
>
1
d
d
x
artanh
x
=
1
1
−
x
2
,
for all real
|
x
|
<
1
d
d
x
arcoth
x
=
1
1
−
x
2
,
for all real
|
x
|
>
1
d
d
x
arsech
x
=
−
1
x
1
−
x
2
,
for all real
x
∈
(
0
,
1
)
d
d
x
arcsch
x
=
−
1
|
x
|
1
+
x
2
,
for all real
x
, except
0
{\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arsinh} x&{}={\frac {1}{\sqrt {x^{2}+1}}},{\text{ for all real }}x\\{\frac {d}{dx}}\operatorname {arcosh} x&{}={\frac {1}{\sqrt {x^{2}-1}}},{\text{ for all real }}x>1\\{\frac {d}{dx}}\operatorname {artanh} x&{}={\frac {1}{1-x^{2}}},{\text{ for all real }}|x|<1\\{\frac {d}{dx}}\operatorname {arcoth} x&{}={\frac {1}{1-x^{2}}},{\text{ for all real }}|x|>1\\{\frac {d}{dx}}\operatorname {arsech} x&{}={\frac {-1}{x{\sqrt {1-x^{2}}}}},{\text{ for all real }}x\in (0,1)\\{\frac {d}{dx}}\operatorname {arcsch} x&{}={\frac {-1}{|x|{\sqrt {1+x^{2}}}}},{\text{ for all real }}x{\text{, except }}0\\\end{aligned}}}
إثبات:
نضع على سبيل المثال θ = arsinh x sinh 2 θ = (sinh θ) 2 ):
d
arsinh
x
d
x
=
d
θ
d
sinh
θ
=
1
cosh
θ
=
1
1
+
sinh
2
θ
=
1
1
+
x
2
.
{\displaystyle {\frac {d\,\operatorname {arsinh} x}{dx}}={\frac {d\theta }{d\sinh \theta }}={\frac {1}{\cosh \theta }}={\frac {1}{\sqrt {1+\sinh ^{2}\theta }}}={\frac {1}{\sqrt {1+x^{2}}}}.}
∫
arsinh
(
x
)
d
x
=
x
arsinh
(
x
)
−
x
2
+
1
+
C
{\displaystyle \int \operatorname {arsinh} (x)\,dx=x\operatorname {arsinh} (x)-{\sqrt {x^{2}+1}}+C}
∫
arcosh
(
x
)
d
x
=
x
arcosh
(
x
)
−
x
2
−
1
+
C
{\displaystyle \int \operatorname {arcosh} (x)\,dx=x\operatorname {arcosh} (x)-{\sqrt {x^{2}-1}}+C}
∫
artanh
(
x
)
d
x
=
x
artanh
(
x
)
+
1
2
ln
(
1
−
x
2
)
+
C
{\displaystyle \int \operatorname {artanh} (x)\,dx=x\operatorname {artanh} (x)+{\frac {1}{2}}\ln \left(1-x^{2}\right)+C}
∫
arcoth
(
x
)
d
x
=
x
arcoth
(
x
)
+
1
2
ln
(
x
2
−
1
)
+
C
{\displaystyle \int \operatorname {arcoth} (x)\,dx=x\operatorname {arcoth} (x)+{\frac {1}{2}}\ln \left(x^{2}-1\right)+C}
∫
arsech
(
x
)
d
x
=
x
arsech
(
x
)
−
2
arctan
1
−
x
1
+
x
+
C
{\displaystyle \int \operatorname {arsech} (x)\,dx=x\operatorname {arsech} (x)-{2}\operatorname {arctan} {\sqrt {\frac {1-x}{1+x}}}+C}
∫
arcsch
(
x
)
d
x
=
x
arcsch
(
x
)
+
arcoth
1
x
2
+
1
+
C
{\displaystyle \int \operatorname {arcsch} (x)\,dx=x\operatorname {arcsch} (x)+\operatorname {arcoth} {\sqrt {{\frac {1}{x^{2}}}+1}}+C}
يمكننا التعبير عن الدوال بواسطة المتسلسلات التالية:
arsinh
x
=
x
−
(
1
2
)
x
3
3
+
(
1
⋅
3
2
⋅
4
)
x
5
5
−
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
7
7
±
⋯
=
∑
n
=
0
∞
(
(
−
1
)
n
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
2
n
+
1
2
n
+
1
,
|
x
|
<
1
{\displaystyle {\begin{aligned}\operatorname {arsinh} x&=x-\left({\frac {1}{2}}\right){\frac {x^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{5}}{5}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{7}}{7}}\pm \cdots \\&=\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{2n+1}}{2n+1}},\qquad \left|x\right|<1\end{aligned}}}
arcosh
x
=
ln
(
2
x
)
−
(
(
1
2
)
x
−
2
2
+
(
1
⋅
3
2
⋅
4
)
x
−
4
4
+
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
−
6
6
+
⋯
)
=
ln
(
2
x
)
−
∑
n
=
1
∞
(
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
−
2
n
2
n
,
|
x
|
>
1
{\displaystyle {\begin{aligned}\operatorname {arcosh} x&=\ln(2x)-\left(\left({\frac {1}{2}}\right){\frac {x^{-2}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{-4}}{4}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{-6}}{6}}+\cdots \right)\\&=\ln(2x)-\sum _{n=1}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-2n}}{2n}},\qquad \left|x\right|>1\end{aligned}}}
artanh
x
=
x
+
x
3
3
+
x
5
5
+
x
7
7
+
⋯
=
∑
n
=
0
∞
x
2
n
+
1
2
n
+
1
,
|
x
|
<
1
{\displaystyle {\begin{aligned}\operatorname {artanh} 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}},\qquad \left|x\right|<1\end{aligned}}}
arcsch
x
=
arsinh
1
x
=
x
−
1
−
(
1
2
)
x
−
3
3
+
(
1
⋅
3
2
⋅
4
)
x
−
5
5
−
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
−
7
7
±
⋯
=
∑
n
=
0
∞
(
(
−
1
)
n
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
−
(
2
n
+
1
)
2
n
+
1
,
|
x
|
>
1
{\displaystyle {\begin{aligned}\operatorname {arcsch} x=\operatorname {arsinh} {\frac {1}{x}}&=x^{-1}-\left({\frac {1}{2}}\right){\frac {x^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{-5}}{5}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{-7}}{7}}\pm \cdots \\&=\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-(2n+1)}}{2n+1}},\qquad \left|x\right|>1\end{aligned}}}
arsech
x
=
arcosh
1
x
=
ln
2
x
−
(
(
1
2
)
x
2
2
+
(
1
⋅
3
2
⋅
4
)
x
4
4
+
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
6
6
+
⋯
)
=
ln
2
x
−
∑
n
=
1
∞
(
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
2
n
2
n
,
0
<
x
≤
1
{\displaystyle {\begin{aligned}\operatorname {arsech} x=\operatorname {arcosh} {\frac {1}{x}}&=\ln {\frac {2}{x}}-\left(\left({\frac {1}{2}}\right){\frac {x^{2}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{4}}{4}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{6}}{6}}+\cdots \right)\\&=\ln {\frac {2}{x}}-\sum _{n=1}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{2n}}{2n}},\qquad 0<x\leq 1\end{aligned}}}
arcoth
x
=
artanh
1
x
=
x
−
1
+
x
−
3
3
+
x
−
5
5
+
x
−
7
7
+
⋯
=
∑
n
=
0
∞
x
−
(
2
n
+
1
)
2
n
+
1
,
|
x
|
>
1
{\displaystyle {\begin{aligned}\operatorname {arcoth} x=\operatorname {artanh} {\frac {1}{x}}&=x^{-1}+{\frac {x^{-3}}{3}}+{\frac {x^{-5}}{5}}+{\frac {x^{-7}}{7}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {x^{-(2n+1)}}{2n+1}},\qquad \left|x\right|>1\end{aligned}}}
^ ترجمة افتراضية من الإنجليزية Area functions .
^ Bronshtein, Ilja N.; Semendyayev, Konstantin A.; Musiol, Gerhard; Mühlig, Heiner (2007). "Chapter 2.10: Area Functions". Handbook of Mathematics سبرنجر . p. 91. doi :10.1007/978-3-540-72122-2 . ISBN 3-540-72121-5 ^ Ebner, Dieter (2005-07-25). Preparatory Course in Mathematics (PDF) (6 ed.). Department of Physics, جامعة كونستانز . Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26 . نسخة محفوظة 26 يوليو 2017 على موقع واي باك مشين .^ Mejlbro, Leif (2006). Real Functions in One Variable – Calculus (PDF) . 1a (1 ed.). Ventus Publishing ApS / Bookboon . ISBN 87-7681-117-4 Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26 . نسخة محفوظة 26 يوليو 2017 على موقع واي باك مشين .^ Mejlbro, Leif (2008). The Argument Principle and Many-valued Functions - Complex Functions Examples (PDF) . c-9 (1 ed.). Ventus Publishing ApS / Bookboon . ISBN 978-87-7681-395-6 Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26 . نسخة محفوظة 10 نوفمبر 2019 على موقع واي باك مشين .^ Mejlbro, Leif (2010-11-11). Stability, Riemann Surfaces, Conformal Mappings - Complex Functions Theory (PDF) . a-3 (1 ed.). Ventus Publishing ApS / Bookboon . ISBN 978-87-7681-702-2 ISBN 87-7681-702-4 . Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26 . نسخة محفوظة 26 يوليو 2017 على موقع واي باك مشين .^ Durán, Mario (2012). Mathematical methods for wave propagation in science and engineering . 1: Fundamentals (1 ed.). Ediciones UC. p. 89. ISBN 978-956141314-6 ISBN 956141314-0 . "نسخة مؤرشفة" . مؤرشف من الأصل في 2020-05-08. اطلع عليه بتاريخ 2019-12-06 .{{استشهاد ويب }}: صيانة الاستشهاد: BOT: original URL status unknown (link )^ Weltner, Klaus; John, Sebastian; Weber, Wolfgang J.; Schuster, Peter; Grosjean, Jean (2014-06-27) [2009]. Mathematics for Physicists and Engineers: Fundamentals and Interactive Study Guide سبرنجر . ISBN 978-364254124-7 ISBN 3642541240 . "نسخة مؤرشفة" . مؤرشف من الأصل في 2020-05-08. اطلع عليه بتاريخ 2019-12-06 .{{استشهاد ويب }}: صيانة الاستشهاد: BOT: original URL status unknown (link )^ Detlef Reimers http://tug.ctan.org/macros/latex/contrib/lapdf/fplot.pdf نسخة محفوظة 2019-11-10 على موقع واي باك مشين .
^ Bacon، Harold Maile (1942). Differential and Integral Calculus الأصل في 2014-07-26.
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