DLMF:10.22.E17 (Q3391): Difference between revisions

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Property / constraint
 

- 1 2 < ℜ ⁡ ν < 1 2 1 2 𝜈 1 2 {\displaystyle{\displaystyle-\tfrac{1}{2}<\Re\nu<\tfrac{1}{2}}}

-\tfrac{1}{2}<\realpart@@{\nu}<\tfrac{1}{2}
Property / constraint: - 1 2 < ℜ ⁡ ν < 1 2 1 2 𝜈 1 2 {\displaystyle{\displaystyle-\tfrac{1}{2}<\Re\nu<\tfrac{1}{2}}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Bessel function of the first kind / rank
 
Normal rank
Property / Symbols used: Bessel function of the first kind / qualifier
 
test:

J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}

\BesselJ{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the first kind / qualifier
 
xml-id: C10.S2.E2.m2aidec
Property / Symbols used
 
Property / Symbols used: Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: Bessel function of the second kind / qualifier
 
test:

Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}

\BesselY{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the second kind / qualifier
 
xml-id: C10.S2.E3.m2adec
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.m2aedec
Property / Symbols used
 
Property / Symbols used: cosecant function / rank
 
Normal rank
Property / Symbols used: cosecant function / qualifier
 
test:

csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}

\csc@@{\NVar{z}}
Property / Symbols used: cosecant function / qualifier
 
xml-id: C4.S14.E5.m2adec
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
test:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2acdec
Property / Symbols used
 
Property / Symbols used: cotangent function / rank
 
Normal rank
Property / Symbols used: cotangent function / qualifier
 
test:

cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}

\cot@@{\NVar{z}}
Property / Symbols used: cotangent function / qualifier
 
xml-id: C4.S14.E7.m2adec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1apdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3apdec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aedec
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.m1kdec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1jdec

Latest revision as of 01:12, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:10.22.E17
No description defined

    Statements

    test
    ∫ 0 1 2 ⁢ π Y 2 ⁢ ν ⁡ ( 2 ⁢ z ⁢ cos ⁡ θ ) ⁢ cos ⁡ ( 2 ⁢ μ ⁢ θ ) ⁢ d θ = 1 2 ⁢ π ⁢ cot ⁡ ( 2 ⁢ ν ⁢ π ) ⁢ J ν + μ ⁡ ( z ) ⁢ J ν - μ ⁡ ( z ) - 1 2 ⁢ π ⁢ csc ⁡ ( 2 ⁢ ν ⁢ π ) ⁢ J μ - ν ⁡ ( z ) ⁢ J - μ - ν ⁡ ( z ) , superscript subscript 0 1 2 𝜋 Bessel-Y-Weber 2 𝜈 2 𝑧 𝜃 2 𝜇 𝜃 𝜃 1 2 𝜋 2 𝜈 𝜋 Bessel-J 𝜈 𝜇 𝑧 Bessel-J 𝜈 𝜇 𝑧 1 2 𝜋 2 𝜈 𝜋 Bessel-J 𝜇 𝜈 𝑧 Bessel-J 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}Y_{2\nu}\left(2z\cos% \theta\right)\cos\left(2\mu\theta\right)\mathrm{d}\theta=\tfrac{1}{2}\pi\cot% \left(2\nu\pi\right)J_{\nu+\mu}\left(z\right)J_{\nu-\mu}\left(z\right)-\tfrac{% 1}{2}\pi\csc\left(2\nu\pi\right)J_{\mu-\nu}\left(z\right)J_{-\mu-\nu}\left(z% \right),}}
    0 references
    DLMF id
    DLMF:10.22.E17
    0 references
    constraint
    - 1 2 < ℜ ⁡ ν < 1 2 1 2 𝜈 1 2 {\displaystyle{\displaystyle-\tfrac{1}{2}<\Re\nu<\tfrac{1}{2}}}
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aidec
    0 references
    Bessel function of the second kind
    test
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    cosecant function
    test
    csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}
    xml-id
    C4.S14.E5.m2adec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    cotangent function
    test
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1apdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3apdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aedec
    0 references
    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1kdec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1jdec
    0 references
     
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