DLMF:19.5.E3 (Q6132): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10759 / rank
 
Normal rank
Property / Symbols used: Q10759 / qualifier
 
test:

( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}

\Pochhammersym{\NVar{a}}{\NVar{n}}
Property / Symbols used: Q10759 / qualifier
 
xml-id: C5.S2.SS3.m1abdec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11819 / rank
 
Normal rank
Property / Symbols used: Q11819 / qualifier
 
test:

D ⁡ ( k ) complete-elliptic-integral-D 𝑘 {\displaystyle{\displaystyle D\left(\NVar{k}\right)}}

\compellintDk@{\NVar{k}}
Property / Symbols used: Q11819 / qualifier
 
xml-id: C19.S2.E8.m3adec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5abdec
Property / Symbols used
 
Property / Symbols used: Q11818 / rank
 
Normal rank
Property / Symbols used: Q11818 / qualifier
 
test:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11818 / qualifier
 
xml-id: C19.S1.XMD2.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11808 / rank
 
Normal rank
Property / Symbols used: Q11808 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11808 / qualifier
 
xml-id: C19.S1.XMD5.m1bdec

Latest revision as of 12:39, 2 January 2020

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Language Label Description Also known as
English
DLMF:19.5.E3
No description defined

    Statements

    test
    D ⁡ ( k ) = π 4 ⁢ ∑ m = 0 ∞ ( 3 2 ) m ⁢ ( 1 2 ) m ( m + 1 ) ! ⁢ m ! ⁢ k 2 ⁢ m = π 4 ⁢ F 1 2 ⁡ ( 3 2 , 1 2 ; 2 ; k 2 ) , complete-elliptic-integral-D 𝑘 𝜋 4 superscript subscript 𝑚 0 Pochhammer 3 2 𝑚 Pochhammer 1 2 𝑚 𝑚 1 𝑚 superscript 𝑘 2 𝑚 𝜋 4 Gauss-hypergeometric-F-as-2F1 3 2 1 2 2 superscript 𝑘 2 {\displaystyle{\displaystyle D\left(k\right)=\frac{\pi}{4}\sum_{m=0}^{\infty}% \frac{{\left(\tfrac{3}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;% m!}k^{2m}=\frac{\pi}{4}{{}_{2}F_{1}}\left(\tfrac{3}{2},\tfrac{1}{2};2;k^{2}% \right),}}
    0 references
    DLMF id
    DLMF:19.5.E3
    0 references
    Symbols used
    Q11581
    test
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5abdec
    0 references
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q11819
    test
    D ⁡ ( k ) complete-elliptic-integral-D 𝑘 {\displaystyle{\displaystyle D\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m3adec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    Q11818
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C19.S1.XMD2.m1bdec
    0 references
    Q11808
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1bdec
    0 references
     
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