DLMF:10.50.E4 (Q3734): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11169 / rank
 
Normal rank
Property / Symbols used: Q11169 / qualifier
 
test:

⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}

\floor{\NVar{x}}
Property / Symbols used: Q11169 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r17.m4adec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2adec
Property / Symbols used
 
Property / Symbols used: spherical Bessel function of the first kind / rank
 
Normal rank
Property / Symbols used: spherical Bessel function of the first kind / qualifier
 
test:

𝗃 n ⁡ ( z ) spherical-Bessel-J 𝑛 𝑧 {\displaystyle{\displaystyle\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)}}

\sphBesselJ{\NVar{n}}@{\NVar{z}}
Property / Symbols used: spherical Bessel function of the first kind / qualifier
 
xml-id: C10.S47.E3.m2abdec
Property / Symbols used
 
Property / Symbols used: spherical Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: spherical Bessel function of the second kind / qualifier
 
test:

𝗒 n ⁡ ( z ) spherical-Bessel-Y 𝑛 𝑧 {\displaystyle{\displaystyle\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)}}

\sphBesselY{\NVar{n}}@{\NVar{z}}
Property / Symbols used: spherical Bessel function of the second kind / qualifier
 
xml-id: C10.S47.E4.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11429 / rank
 
Normal rank
Property / Symbols used: Q11429 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11429 / qualifier
 
xml-id: C10.S1.XMD2.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11428 / rank
 
Normal rank
Property / Symbols used: Q11428 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11428 / qualifier
 
xml-id: C10.S1.XMD3.m1dec
Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1cdec
Property / Symbols used
 
Property / Symbols used: expansion (locally) / rank
 
Normal rank
Property / Symbols used: expansion (locally) / qualifier
 
test:

a k ⁢ ( ν ) subscript 𝑎 𝑘 𝜈 {\displaystyle{\displaystyle a_{k}(\nu)}}

a_{k}(\nu)
Property / Symbols used: expansion (locally) / qualifier
 
xml-id: C10.S17.XMD1.m1dec

Latest revision as of 14:12, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:10.50.E4
No description defined

    Statements

    test
    𝗃 0 ⁡ ( z ) ⁢ 𝗃 n ⁡ ( z ) + 𝗒 0 ⁡ ( z ) ⁢ 𝗒 n ⁡ ( z ) = cos ⁡ ( 1 2 ⁢ n ⁢ π ) ⁢ ∑ k = 0 ⌊ n / 2 ⌋ ( - 1 ) k ⁢ a 2 ⁢ k ⁢ ( n + 1 2 ) z 2 ⁢ k + 2 + sin ⁡ ( 1 2 ⁢ n ⁢ π ) ⁢ ∑ k = 0 ⌊ ( n - 1 ) / 2 ⌋ ( - 1 ) k ⁢ a 2 ⁢ k + 1 ⁢ ( n + 1 2 ) z 2 ⁢ k + 3 , spherical-Bessel-J 0 𝑧 spherical-Bessel-J 𝑛 𝑧 spherical-Bessel-Y 0 𝑧 spherical-Bessel-Y 𝑛 𝑧 1 2 𝑛 𝜋 superscript subscript 𝑘 0 𝑛 2 superscript 1 𝑘 subscript 𝑎 2 𝑘 𝑛 1 2 superscript 𝑧 2 𝑘 2 1 2 𝑛 𝜋 superscript subscript 𝑘 0 𝑛 1 2 superscript 1 𝑘 subscript 𝑎 2 𝑘 1 𝑛 1 2 superscript 𝑧 2 𝑘 3 {\displaystyle{\displaystyle\mathsf{j}_{0}\left(z\right)\mathsf{j}_{n}\left(z% \right)+\mathsf{y}_{0}\left(z\right)\mathsf{y}_{n}\left(z\right)=\cos\left(% \tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor n/2\right\rfloor}(-1)^{k}\frac% {a_{2k}(n+\tfrac{1}{2})}{z^{2k+2}}+\sin\left(\tfrac{1}{2}n\pi\right)\sum_{k=0}% ^{\left\lfloor(n-1)/2\right\rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{% 2k+3}},}}
    0 references
    DLMF id
    DLMF:10.50.E4
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q11169
    test
    ⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}
    xml-id
    introduction.Sx4.p1.t1.r17.m4adec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    spherical Bessel function of the first kind
    test
    𝗃 n ⁡ ( z ) spherical-Bessel-J 𝑛 𝑧 {\displaystyle{\displaystyle\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)}}
    xml-id
    C10.S47.E3.m2abdec
    0 references
    spherical Bessel function of the second kind
    test
    𝗒 n ⁡ ( z ) spherical-Bessel-Y 𝑛 𝑧 {\displaystyle{\displaystyle\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)}}
    xml-id
    C10.S47.E4.m2abdec
    0 references
    Q11429
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C10.S1.XMD2.m1cdec
    0 references
    Q11428
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1dec
    0 references
    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1cdec
    0 references
    expansion (locally)
    test
    a k ⁢ ( ν ) subscript 𝑎 𝑘 𝜈 {\displaystyle{\displaystyle a_{k}(\nu)}}
    xml-id
    C10.S17.XMD1.m1dec
    0 references
     
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