1.15.E40 (Q552): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: function / rank
 
Normal rank
Property / Symbols used: function / qualifier
 
DLMF defining formula:

Φ ⁢ ( z ) Φ 𝑧 {\displaystyle{\displaystyle\Phi(z)}}

\Phi(z)
Property / Symbols used: function / qualifier
 
xml-id: C1.S15.XMD9.m1adec

Revision as of 09:34, 21 December 2021

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1.15.E40
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    Statements

    DLMF defining formula
    ℑ ⁡ Φ ⁢ ( x + i ⁢ y ) = 1 π ⁢ ∫ - ∞ ∞ f ⁢ ( t ) ⁢ x - t ( x - t ) 2 + y 2 ⁢ d t Φ 𝑥 𝑖 𝑦 1 𝜋 subscript superscript 𝑓 𝑡 𝑥 𝑡 superscript 𝑥 𝑡 2 superscript 𝑦 2 𝑡 {\displaystyle{\displaystyle\Im\Phi(x+iy)=\frac{1}{\pi}\int^{\infty}_{-\infty}% f(t)\frac{x-t}{(x-t)^{2}+y^{2}}\mathrm{d}t}}
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aodec
    0 references
    imaginary part
    DLMF defining formula
    ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}
    xml-id
    C1.S9.E2.m2adec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ajdec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aodec
    0 references
    function
    DLMF defining formula
    Φ ⁢ ( z ) Φ 𝑧 {\displaystyle{\displaystyle\Phi(z)}}
    xml-id
    C1.S15.XMD9.m1adec
    0 references

    Identifiers

     
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