DLMF:12.9.E4 (Q4153): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10759 / rank
 
Normal rank
Property / Symbols used: Q10759 / qualifier
 
Defining formula:

( 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.m1acdec
Property / Symbols used
 
Property / Symbols used: Q10931 / rank
 
Normal rank
Property / Symbols used: Q10931 / qualifier
 
Defining formula:

∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}

\asympexp
Property / Symbols used: Q10931 / qualifier
 
xml-id: C2.S1.SS3.p1.m11acdec
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
 
Defining formula:

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.m2acdec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2acdec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
Defining formula:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5acdec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11517 / rank
 
Normal rank
Property / Symbols used: Q11517 / qualifier
 
Defining formula:

V ⁡ ( a , z ) parabolic-V 𝑎 𝑧 {\displaystyle{\displaystyle V\left(\NVar{a},\NVar{z}\right)}}

\paraV@{\NVar{a}}{\NVar{z}}
Property / Symbols used: Q11517 / qualifier
 
xml-id: C12.S2.SS1.p1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
Defining formula:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1acdec
Property / Symbols used
 
Property / Symbols used: Q11513 / rank
 
Normal rank
Property / Symbols used: Q11513 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11513 / qualifier
 
xml-id: C12.S1.XMD3.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11526 / rank
 
Normal rank
Property / Symbols used: Q11526 / qualifier
 
Defining formula:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q11526 / qualifier
 
xml-id: C12.S1.XMD5.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11516 / rank
 
Normal rank
Property / Symbols used: Q11516 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11516 / qualifier
 
xml-id: C12.S1.XMD6.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11527 / rank
 
Normal rank
Property / Symbols used: Q11527 / qualifier
 
Defining formula:

δ 𝛿 {\displaystyle{\displaystyle\delta}}

\delta
Property / Symbols used: Q11527 / qualifier
 
xml-id: C12.S1.XMD8.m1cdec

Latest revision as of 15:16, 2 January 2020

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DLMF:12.9.E4
No description defined

    Statements

    Defining formula
    V ⁡ ( a , z ) ∼ 2 π ⁢ e 1 4 ⁢ z 2 ⁢ z a - 1 2 ⁢ ∑ s = 0 ∞ ( 1 2 - a ) 2 ⁢ s s ! ⁢ ( 2 ⁢ z 2 ) s ± i Γ ⁡ ( 1 2 - a ) ⁢ e - 1 4 ⁢ z 2 ⁢ z - a - 1 2 ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ ( 1 2 + a ) 2 ⁢ s s ! ⁢ ( 2 ⁢ z 2 ) s , asymptotic-expansion parabolic-V 𝑎 𝑧 plus-or-minus 2 𝜋 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑧 𝑎 1 2 superscript subscript 𝑠 0 Pochhammer 1 2 𝑎 2 𝑠 𝑠 superscript 2 superscript 𝑧 2 𝑠 𝑖 Euler-Gamma 1 2 𝑎 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑧 𝑎 1 2 superscript subscript 𝑠 0 superscript 1 𝑠 Pochhammer 1 2 𝑎 2 𝑠 𝑠 superscript 2 superscript 𝑧 2 𝑠 {\displaystyle{\displaystyle V\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{\frac% {1}{4}z^{2}}z^{a-\frac{1}{2}}\sum_{s=0}^{\infty}\frac{{\left(\tfrac{1}{2}-a% \right)_{2s}}}{s!(2z^{2})^{s}}\pm\frac{i}{\Gamma\left(\tfrac{1}{2}-a\right)}e^% {-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(% \tfrac{1}{2}+a\right)_{2s}}}{s!(2z^{2})^{s}},}}
    0 references
    DLMF id
    DLMF:12.9.E4
    0 references
    constraint
    - 1 4 ⁢ π + δ ≤ ± ph ⁡ z ≤ 3 4 ⁢ π - δ 1 4 𝜋 𝛿 plus-or-minus phase 𝑧 3 4 𝜋 𝛿 {\displaystyle{\displaystyle-\tfrac{1}{4}\pi+\delta\leq\pm\operatorname{ph}z% \leq\tfrac{3}{4}\pi-\delta}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1acdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q11517
    Defining formula
    V ⁡ ( a , z ) parabolic-V 𝑎 𝑧 {\displaystyle{\displaystyle V\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m2aadec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1acdec
    0 references
    Q11513
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C12.S1.XMD3.m1cdec
    0 references
    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1cdec
    0 references
    Q11516
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C12.S1.XMD6.m1cdec
    0 references
    Q11527
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C12.S1.XMD8.m1cdec
    0 references
     
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