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DLMF:13.9.E16 - MaRDI portal

DLMF:13.9.E16 (Q5156)

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DLMF:13.9.E16
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    Statements

    DLMF defining formula
    a = - n - 2 π ⁢ z ⁢ n - 2 ⁢ z π 2 + 1 2 ⁢ b + 1 4 + z 2 ⁢ ( 1 3 - 4 ⁢ π - 2 ) + z - ( b - 1 ) 2 + 1 4 4 ⁢ π ⁢ z ⁢ n + O ⁡ ( 1 n ) , 𝑎 𝑛 2 𝜋 𝑧 𝑛 2 𝑧 superscript 𝜋 2 1 2 𝑏 1 4 superscript 𝑧 2 1 3 4 superscript 𝜋 2 𝑧 superscript 𝑏 1 2 1 4 4 𝜋 𝑧 𝑛 Big-O 1 𝑛 {\displaystyle{\displaystyle a=-n-\frac{2}{\pi}\sqrt{zn}-\frac{2z}{\pi^{2}}+% \tfrac{1}{2}b+\tfrac{1}{4}+\frac{z^{2}\left(\frac{1}{3}-4\pi^{-2}\right)+z-(b-% 1)^{2}+\frac{1}{4}}{4\pi\sqrt{zn}}+O\left(\frac{1}{n}\right),}}
    0 references
    Symbols used
    order not exceeding
    DLMF defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2acdec
    0 references
    Q10905
    DLMF defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    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
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1bdec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1bdec
    0 references
     
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