DLMF:12.9.E3 (Q4849)

From MaRDI portal
No description defined
Language Label Description Also known as
English
DLMF:12.9.E3
No description defined

    Statements

    DLMF defining formula
    U ⁡ ( a , z ) ∼ e - 1 4 ⁢ z 2 ⁢ z - a - 1 2 ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ ( 1 2 + a ) 2 ⁢ s s ! ⁢ ( 2 ⁢ z 2 ) s ± i ⁢ 2 ⁢ π Γ ⁡ ( 1 2 + a ) ⁢ e ∓ i ⁢ π ⁢ a ⁢ e 1 4 ⁢ z 2 ⁢ z a - 1 2 ⁢ ∑ s = 0 ∞ ( 1 2 - a ) 2 ⁢ s s ! ⁢ ( 2 ⁢ z 2 ) s , asymptotic-expansion parabolic-U 𝑎 𝑧 plus-or-minus 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 𝑖 2 𝜋 Euler-Gamma 1 2 𝑎 superscript 𝑒 minus-or-plus 𝑖 𝜋 𝑎 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 {\displaystyle{\displaystyle U\left(a,z\right)\sim e^{-\frac{1}{4}z^{2}}z^{-a-% \frac{1}{2}}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(\frac{1}{2}+a\right)_{2s}}% }{s!(2z^{2})^{s}}\pm i\frac{\sqrt{2\pi}}{\Gamma\left(\tfrac{1}{2}+a\right)}e^{% \mp i\pi a}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}},}}
    0 references
    constraint
    1 4 ⁢ π + δ ≤ ± ph ⁡ z ≤ 5 4 ⁢ π - δ 1 4 𝜋 𝛿 plus-or-minus phase 𝑧 5 4 𝜋 𝛿 {\displaystyle{\displaystyle\tfrac{1}{4}\pi+\delta\leq\pm\operatorname{ph}z% \leq\tfrac{5}{4}\pi-\delta}}
    0 references
    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q10733
    DLMF defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
    0 references
    Q10905
    DLMF defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q11291
    DLMF defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1aadec
    0 references
    phase
    DLMF defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1abdec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C12.S1.XMD3.m1bdec
    0 references
    nonnegative integer
    DLMF defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1bdec
    0 references
    real or complex parameter
    DLMF defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C12.S1.XMD6.m1bdec
    0 references
    arbitrary small positive constant
    DLMF defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C12.S1.XMD8.m1bdec
    0 references

    Identifiers

     
    Retrieved from "https://portal.mardi4nfdi.de/w/index.php?title=Item:Q4849&oldid=24173532"
    Powered by MediaWiki