DLMF:7.14.E2 (Q2434): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
Defining formula:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1aadec
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.m1adec

Latest revision as of 01:51, 2 January 2020

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DLMF:7.14.E2
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    Statements

    Defining formula
    ∫ 0 ∞ e - a ⁢ t ⁢ erf ⁡ ( b ⁢ t ) ⁢ d t = 1 a ⁢ e a 2 / ( 4 ⁢ b 2 ) ⁢ erfc ⁡ ( a 2 ⁢ b ) , superscript subscript 0 superscript 𝑒 𝑎 𝑡 error-function 𝑏 𝑡 𝑡 1 𝑎 superscript 𝑒 superscript 𝑎 2 4 superscript 𝑏 2 complementary-error-function 𝑎 2 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt% \right)\mathrm{d}t=\frac{1}{a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac% {a}{2b}\right),}}
    0 references
    DLMF id
    DLMF:7.14.E2
    0 references
    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    0 references
    | ph ⁡ b | < 1 4 ⁢ π phase 𝑏 1 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}b|<\tfrac{1}{4}\pi}}
    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.m2aadec
    0 references
    complementary error function
    Defining formula
    erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}
    xml-id
    C7.S2.E2.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    error function
    Defining formula
    erf ⁡ z error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erf}\NVar{z}}}
    xml-id
    C7.S2.E1.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aadec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
     
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