DLMF:2.4.E7 (Q760): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aedec
Property / Symbols used
 
Property / Symbols used: Q10973 / rank
 
Normal rank
Property / Symbols used: Q10973 / qualifier
 
Defining formula:

q ⁢ ( t ) 𝑞 𝑡 {\displaystyle{\displaystyle q(t)}}

q(t)
Property / Symbols used: Q10973 / qualifier
 
xml-id: C2.S4.XMD16.m1ddec
Property / Symbols used
 
Property / Symbols used: Q10975 / rank
 
Normal rank
Property / Symbols used: Q10975 / qualifier
 
Defining formula:

Q ⁢ ( z ) 𝑄 𝑧 {\displaystyle{\displaystyle Q(z)}}

Q(z)
Property / Symbols used: Q10975 / qualifier
 
xml-id: C2.S4.XMD6.m1ddec
Property / Symbols used
 
Property / Symbols used: Q10976 / rank
 
Normal rank
Property / Symbols used: Q10976 / qualifier
 
Defining formula:

σ 𝜎 {\displaystyle{\displaystyle\sigma}}

\sigma
Property / Symbols used: Q10976 / qualifier
 
xml-id: C2.S4.XMD7.m1cdec
Property / Symbols used
 
Property / Symbols used: Q10977 / rank
 
Normal rank
Property / Symbols used: Q10977 / qualifier
 
Defining formula:

F ⁢ ( z ) 𝐹 𝑧 {\displaystyle{\displaystyle F(z)}}

F(z)
Property / Symbols used: Q10977 / qualifier
 
xml-id: C2.S4.XMD8.m1adec
Property / Symbols used
 
Property / Symbols used: Q10978 / rank
 
Normal rank
Property / Symbols used: Q10978 / qualifier
 
Defining formula:

f ⁢ ( x ) 𝑓 𝑥 {\displaystyle{\displaystyle f(x)}}

f(x)
Property / Symbols used: Q10978 / qualifier
 
xml-id: C2.S4.XMD9.m1adec

Latest revision as of 16:58, 1 January 2020

No description defined
Language Label Description Also known as
English
DLMF:2.4.E7
No description defined

    Statements

    Defining formula
    q ⁢ ( t ) - f ⁢ ( t ) = e σ ⁢ t 2 ⁢ π ⁢ lim η → ∞ ⁡ ∫ - η η e i ⁢ t ⁢ τ ⁢ ( Q ⁢ ( σ + i ⁢ τ ) - F ⁢ ( σ + i ⁢ τ ) ) ⁢ d τ . 𝑞 𝑡 𝑓 𝑡 superscript 𝑒 𝜎 𝑡 2 𝜋 subscript → 𝜂 superscript subscript 𝜂 𝜂 superscript 𝑒 𝑖 𝑡 𝜏 𝑄 𝜎 𝑖 𝜏 𝐹 𝜎 𝑖 𝜏 𝜏 {\displaystyle{\displaystyle q(t)-f(t)=\frac{e^{\sigma t}}{2\pi}\lim\limits_{% \eta\to\infty}\int_{-\eta}^{\eta}e^{it\tau}(Q(\sigma+i\tau)-F(\sigma+i\tau))% \mathrm{d}\tau.}}
    0 references
    DLMF id
    DLMF:2.4.E7
    0 references
    Symbols used
    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
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    Q10973
    Defining formula
    q ⁢ ( t ) 𝑞 𝑡 {\displaystyle{\displaystyle q(t)}}
    xml-id
    C2.S4.XMD16.m1ddec
    0 references
    Q10975
    Defining formula
    Q ⁢ ( z ) 𝑄 𝑧 {\displaystyle{\displaystyle Q(z)}}
    xml-id
    C2.S4.XMD6.m1ddec
    0 references
    Q10976
    Defining formula
    σ 𝜎 {\displaystyle{\displaystyle\sigma}}
    xml-id
    C2.S4.XMD7.m1cdec
    0 references
    Q10977
    Defining formula
    F ⁢ ( z ) 𝐹 𝑧 {\displaystyle{\displaystyle F(z)}}
    xml-id
    C2.S4.XMD8.m1adec
    0 references
    Q10978
    Defining formula
    f ⁢ ( x ) 𝑓 𝑥 {\displaystyle{\displaystyle f(x)}}
    xml-id
    C2.S4.XMD9.m1adec
    0 references
     
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