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DLMF:7.6.E5 - MaRDI portal

DLMF:7.6.E5 (Q3062)

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DLMF:7.6.E5
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    Statements

    DLMF defining formula
    C ⁡ ( z ) = cos ⁡ ( 1 2 ⁢ π ⁢ z 2 ) ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ π 2 ⁢ n 1 ⋅ 3 ⁢ ⋯ ⁢ ( 4 ⁢ n + 1 ) ⁢ z 4 ⁢ n + 1 + sin ⁡ ( 1 2 ⁢ π ⁢ z 2 ) ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ π 2 ⁢ n + 1 1 ⋅ 3 ⁢ ⋯ ⁢ ( 4 ⁢ n + 3 ) ⁢ z 4 ⁢ n + 3 . Fresnel-cosine-integral 𝑧 1 2 𝜋 superscript 𝑧 2 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝜋 2 𝑛 ⋅ 1 3 ⋯ 4 𝑛 1 superscript 𝑧 4 𝑛 1 1 2 𝜋 superscript 𝑧 2 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝜋 2 𝑛 1 ⋅ 1 3 ⋯ 4 𝑛 3 superscript 𝑧 4 𝑛 3 {\displaystyle{\displaystyle C\left(z\right)=\cos\left(\tfrac{1}{2}\pi z^{2}% \right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1)}z^{4n+1% }+\sin\left(\tfrac{1}{2}\pi z^{2}\right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{% 2n+1}}{1\cdot 3\cdots(4n+3)}z^{4n+3}.}}
    0 references
    Symbols used
    Fresnel integral
    DLMF defining formula
    C ⁡ ( z ) Fresnel-cosine-integral 𝑧 {\displaystyle{\displaystyle C\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E7.m2aadec
    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.m2acdec
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    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    sine function
    DLMF defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1ddec
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    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C7.S1.XMD3.m1ddec
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    Identifiers

     
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