DLMF:7.7.E6 (Q2376): Difference between revisions

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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.m2aedec
Property / Symbols used
 
Property / Symbols used: complementary error function / rank
 
Normal rank
Property / Symbols used: complementary error function / qualifier
 
Defining formula:

erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}

\erfc@@{\NVar{z}}
Property / Symbols used: complementary error function / qualifier
 
xml-id: C7.S2.E2.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.m1aedec
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.m2aedec
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: 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.m1acdec
Property / Symbols used
 
Property / Symbols used: Q11313 / rank
 
Normal rank
Property / Symbols used: Q11313 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11313 / qualifier
 
xml-id: C7.S1.XMD1.m1dec

Latest revision as of 01:44, 2 January 2020

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English
DLMF:7.7.E6
No description defined

    Statements

    Defining formula
    ∫ x ∞ e - ( a ⁢ t 2 + 2 ⁢ b ⁢ t + c ) ⁢ d t = 1 2 ⁢ π a ⁢ e ( b 2 - a ⁢ c ) / a ⁢ erfc ⁡ ( a ⁢ x + b a ) , superscript subscript 𝑥 superscript 𝑒 𝑎 superscript 𝑡 2 2 𝑏 𝑡 𝑐 𝑡 1 2 𝜋 𝑎 superscript 𝑒 superscript 𝑏 2 𝑎 𝑐 𝑎 complementary-error-function 𝑎 𝑥 𝑏 𝑎 {\displaystyle{\displaystyle\int_{x}^{\infty}e^{-(at^{2}+2bt+c)}\mathrm{d}t=% \frac{1}{2}\sqrt{\frac{\pi}{a}}e^{(b^{2}-ac)/a}\operatorname{erfc}\left(\sqrt{% a}x+\frac{b}{\sqrt{a}}\right),}}
    0 references
    DLMF id
    DLMF:7.7.E6
    0 references
    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    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.m2aedec
    0 references
    complementary error function
    Defining formula
    erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}
    xml-id
    C7.S2.E2.m2abdec
    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
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1acdec
    0 references
    Q11313
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C7.S1.XMD1.m1dec
    0 references
     
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