DLMF:10.8.E2 (Q3064): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1adec

Latest revision as of 15:10, 2 January 2020

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DLMF:10.8.E2
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    Statements

    Defining formula
    Y 0 ⁡ ( z ) = 2 π ⁢ ( ln ⁡ ( 1 2 ⁢ z ) + γ ) ⁢ J 0 ⁡ ( z ) + 2 π ⁢ ( 1 4 ⁢ z 2 ( 1 ! ) 2 - ( 1 + 1 2 ) ⁢ ( 1 4 ⁢ z 2 ) 2 ( 2 ! ) 2 + ( 1 + 1 2 + 1 3 ) ⁢ ( 1 4 ⁢ z 2 ) 3 ( 3 ! ) 2 - ⋯ ) , Bessel-Y-Weber 0 𝑧 2 𝜋 1 2 𝑧 Bessel-J 0 𝑧 2 𝜋 1 4 superscript 𝑧 2 superscript 1 2 1 1 2 superscript 1 4 superscript 𝑧 2 2 superscript 2 2 1 1 2 1 3 superscript 1 4 superscript 𝑧 2 3 superscript 3 2 ⋯ {\displaystyle{\displaystyle Y_{0}\left(z\right)=\frac{2}{\pi}\left(\ln\left(% \tfrac{1}{2}z\right)+\gamma\right)J_{0}\left(z\right)+\frac{2}{\pi}\left(\frac% {\tfrac{1}{4}z^{2}}{(1!)^{2}}-(1+\tfrac{1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(% 2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}% -\cdots\right),}}
    0 references
    DLMF id
    DLMF:10.8.E2
    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.m2aadec
    0 references
    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1adec
    0 references
     
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