DLMF:12.12.E3 (Q4228): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1abdec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2abdec
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.m3abdec
Property / Symbols used
 
Property / Symbols used: Q11320 / rank
 
Normal rank
Property / Symbols used: Q11320 / qualifier
 
Defining formula:

U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}

\paraU@{\NVar{a}}{\NVar{z}}
Property / Symbols used: Q11320 / qualifier
 
xml-id: C12.S2.SS1.p1.m1abdec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1abdec
Property / Symbols used
 
Property / Symbols used: Q11520 / rank
 
Normal rank
Property / Symbols used: Q11520 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11520 / qualifier
 
xml-id: C12.S1.XMD1.m1dec
Property / Symbols used
 
Property / Symbols used: Q11516 / rank
 
Normal rank
Property / Symbols used: Q11516 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11516 / qualifier
 
xml-id: C12.S1.XMD6.m1bdec

Latest revision as of 15:25, 2 January 2020

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DLMF:12.12.E3
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e - 1 4 ⁢ t 2 ⁢ t - a - 1 2 ⁢ ( x 2 + t 2 ) - 1 ⁢ U ⁡ ( a , t ) ⁢ d t = π / 2 ⁢ Γ ⁡ ( 1 2 - a ) ⁢ x - a - 3 2 ⁢ e 1 4 ⁢ x 2 ⁢ U ⁡ ( - a , x ) , superscript subscript 0 superscript 𝑒 1 4 superscript 𝑡 2 superscript 𝑡 𝑎 1 2 superscript superscript 𝑥 2 superscript 𝑡 2 1 parabolic-U 𝑎 𝑡 𝑡 𝜋 2 Euler-Gamma 1 2 𝑎 superscript 𝑥 𝑎 3 2 superscript 𝑒 1 4 superscript 𝑥 2 parabolic-U 𝑎 𝑥 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{% 1}{2}}(x^{2}+t^{2})^{-1}U\left(a,t\right)\mathrm{d}t=\sqrt{\pi/2}\Gamma\left(% \tfrac{1}{2}-a\right)x^{-a-\frac{3}{2}}e^{\frac{1}{4}x^{2}}U\left(-a,x\right),}}
    0 references
    DLMF id
    DLMF:12.12.E3
    0 references
    constraint
    ℜ ⁡ a < 1 2 , x > 0 formulae-sequence 𝑎 1 2 𝑥 0 {\displaystyle{\displaystyle\Re a<\tfrac{1}{2},x>0}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1abdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
    0 references
    Q11520
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C12.S1.XMD1.m1dec
    0 references
    Q11516
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C12.S1.XMD6.m1bdec
    0 references
     
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