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DLMF:15.12.E5
(Q5862)
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English
DLMF:15.12.E5
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Statements
DLMF defining formula
π
β‘
(
a
+
Ξ»
,
b
-
Ξ»
c
;
1
2
-
1
2
β’
z
)
=
2
(
a
+
b
-
1
)
/
2
β’
(
z
+
1
)
(
c
-
a
-
b
-
1
)
/
2
(
z
-
1
)
c
/
2
β’
ΞΆ
β’
sinh
β‘
ΞΆ
β’
(
Ξ»
+
1
2
β’
a
-
1
2
β’
b
)
1
-
c
β’
(
I
c
-
1
β‘
(
(
Ξ»
+
1
2
β’
a
-
1
2
β’
b
)
β’
ΞΆ
)
β’
(
1
+
O
β‘
(
Ξ»
-
2
)
)
+
I
c
-
2
β‘
(
(
Ξ»
+
1
2
β’
a
-
1
2
β’
b
)
β’
ΞΆ
)
2
β’
Ξ»
+
a
-
b
β’
(
(
c
-
1
2
)
β’
(
c
-
3
2
)
β’
(
1
ΞΆ
-
coth
β‘
ΞΆ
)
+
1
2
β’
(
2
β’
c
-
a
-
b
-
1
)
β’
(
a
+
b
-
1
)
β’
tanh
β‘
(
1
2
β’
ΞΆ
)
+
O
β‘
(
Ξ»
-
2
)
)
)
,
scaled-hypergeometric-bold-F
π
π
π
π
π
1
2
1
2
π§
superscript
2
π
π
1
2
superscript
π§
1
π
π
π
1
2
superscript
π§
1
π
2
π
π
superscript
π
1
2
π
1
2
π
1
π
modified-Bessel-first-kind
π
1
π
1
2
π
1
2
π
π
1
Big-O
superscript
π
2
modified-Bessel-first-kind
π
2
π
1
2
π
1
2
π
π
2
π
π
π
π
1
2
π
3
2
1
π
hyperbolic-cotangent
π
1
2
2
π
π
π
1
π
π
1
1
2
π
Big-O
superscript
π
2
{\displaystyle{\displaystyle\mathbf{F}\left({a+\lambda,b-\lambda\atop c};% \tfrac{1}{2}-\tfrac{1}{2}z\right)=2^{(a+b-1)/2}\frac{(z+1)^{(c-a-b-1)/2}}{(z-1% )^{c/2}}\sqrt{\zeta\sinh\zeta}\left(\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b\right)% ^{1-c}\left(I_{c-1}\left((\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)(1+O% (\lambda^{-2}))+\frac{I_{c-2}\left((\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta% \right)}{2\lambda+a-b}\left(\left(c-\tfrac{1}{2}\right)\left(c-\tfrac{3}{2}% \right)\left(\frac{1}{\zeta}-\coth\zeta\right)+\tfrac{1}{2}(2c-a-b-1)(a+b-1)% \tanh\left(\tfrac{1}{2}\zeta\right)+O(\lambda^{-2})\right)\right),}}
0 references
Symbols used
order not exceeding
DLMF defining formula
O
β‘
(
x
)
Big-O
π₯
{\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
xml-id
C2.S1.E3.m2aadec
0 references
hyperbolic cotangent function
DLMF defining formula
coth
β‘
z
hyperbolic-cotangent
π§
{\displaystyle{\displaystyle\coth\NVar{z}}}
xml-id
C4.S28.E7.m2adec
0 references
hyperbolic sine function
DLMF defining formula
sinh
β‘
z
π§
{\displaystyle{\displaystyle\sinh\NVar{z}}}
xml-id
C4.S28.E1.m2adec
0 references
hyperbolic tangent function
DLMF defining formula
tanh
β‘
z
π§
{\displaystyle{\displaystyle\tanh\NVar{z}}}
xml-id
C4.S28.E4.m2adec
0 references
modified Bessel function of the first kind
DLMF defining formula
I
Ξ½
β‘
(
z
)
modified-Bessel-first-kind
π
π§
{\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
xml-id
C10.S25.E2.m2adec
0 references
Olverβs hypergeometric function
DLMF defining formula
π
β‘
(
a
,
b
;
c
;
z
)
scaled-hypergeometric-bold-F
π
π
π
π§
{\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
xml-id
C15.S2.E2.m2adec
0 references
complex variable
DLMF defining formula
z
π§
{\displaystyle{\displaystyle z}}
xml-id
C15.S1.XMD3.m1ddec
0 references
real or complex parameter
DLMF defining formula
a
π
{\displaystyle{\displaystyle a}}
xml-id
C15.S1.XMD4.m1cdec
0 references
real or complex parameter
DLMF defining formula
b
π
{\displaystyle{\displaystyle b}}
xml-id
C15.S1.XMD5.m1cdec
0 references
real or complex parameter
DLMF defining formula
c
π
{\displaystyle{\displaystyle c}}
xml-id
C15.S1.XMD6.m1cdec
0 references
change of variable
DLMF defining formula
ΞΆ
π
{\displaystyle{\displaystyle\zeta}}
xml-id
C15.S12.XMD5.m1dec
0 references
instance of
Digital Library of Mathematical Functions Formula
0 references
MaRDI profile type
MaRDI formula profile
0 references
Identifiers
Digital Library of Mathematical Functions ID
15.12.E5
0 references
Sitelinks
Mathematics
(1 entry)
mardi
Formula:5862
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