DLMF:13.8.E10 (Q4435): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11091 / rank
 
Normal rank
Property / Symbols used: Q11091 / qualifier
 
Defining formula:

env ⁢ Y ν ⁡ ( x ) envelope-Bessel-Y 𝜈 𝑥 {\displaystyle{\displaystyle\mathrm{env}\mskip-2.0mu Y_{\NVar{\nu}}\left(\NVar% {x}\right)}}

\envBesselY{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Q11091 / qualifier
 
xml-id: C2.S8.SS4.p5.m4adec

Revision as of 15:58, 2 January 2020

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DLMF:13.8.E10
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    Statements

    Defining formula
    U ⁡ ( a , b , x ) = Γ ⁡ ( 1 2 ⁢ b - a + 1 2 ) ⁢ e 1 2 ⁢ x ⁢ x 1 2 - 1 2 ⁢ b ⁢ ( cos ⁡ ( a ⁢ π ) ⁢ J b - 1 ⁡ ( 2 ⁢ x ⁢ ( b - 2 ⁢ a ) ) - sin ⁡ ( a ⁢ π ) ⁢ Y b - 1 ⁡ ( 2 ⁢ x ⁢ ( b - 2 ⁢ a ) ) + env ⁢ Y b - 1 ⁡ ( 2 ⁢ x ⁢ ( b - 2 ⁢ a ) ) ⁢ O ⁡ ( | a | - 1 2 ) ) , Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑥 Euler-Gamma 1 2 𝑏 𝑎 1 2 superscript 𝑒 1 2 𝑥 superscript 𝑥 1 2 1 2 𝑏 𝑎 𝜋 Bessel-J 𝑏 1 2 𝑥 𝑏 2 𝑎 𝑎 𝜋 Bessel-Y-Weber 𝑏 1 2 𝑥 𝑏 2 𝑎 envelope-Bessel-Y 𝑏 1 2 𝑥 𝑏 2 𝑎 Big-O 𝑎 1 2 {\displaystyle{\displaystyle U\left(a,b,x\right)=\Gamma\left(\tfrac{1}{2}b-a+% \tfrac{1}{2}\right)e^{\frac{1}{2}x}x^{\frac{1}{2}-\frac{1}{2}b}\*\left(\cos% \left(a\pi\right)J_{b-1}\left(\sqrt{2x(b-2a)}\right)-\sin\left(a\pi\right)Y_{b% -1}\left(\sqrt{2x(b-2a)}\right)+\mathrm{env}\mskip-2.0mu Y_{b-1}\left(\sqrt{2x% (b-2a)}\right)O\left({\left|a\right|^{-\frac{1}{2}}}\right)\right),}}
    0 references
    DLMF id
    DLMF:13.8.E10
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aadec
    0 references
    Bessel function of the second kind
    Defining formula
    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
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aedec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q11091
    Defining formula
    env ⁢ Y ν ⁡ ( x ) envelope-Bessel-Y 𝜈 𝑥 {\displaystyle{\displaystyle\mathrm{env}\mskip-2.0mu Y_{\NVar{\nu}}\left(\NVar% {x}\right)}}
    xml-id
    C2.S8.SS4.p5.m4adec
    0 references
     
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