DLMF:7.14.E1 (Q2433): Difference between revisions

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Property / constraint
 

| ph ⁡ b | < 1 4 ⁢ π phase 𝑏 1 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}b|<\tfrac{1}{4}\pi}}

|\phase@@{b}|<\tfrac{1}{4}\pi
Property / constraint: | ph ⁡ b | < 1 4 ⁢ π phase 𝑏 1 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}b|<\tfrac{1}{4}\pi}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Dawson’s integral / rank
 
Normal rank
Property / Symbols used: Dawson’s integral / qualifier
 
Defining formula:

F ⁡ ( z ) Dawsons-integral 𝑧 {\displaystyle{\displaystyle F\left(\NVar{z}\right)}}

\DawsonsintF@{\NVar{z}}
Property / Symbols used: Dawson’s integral / qualifier
 
xml-id: C7.S2.E5.m2adec
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.m2adec
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.m2adec
Property / Symbols used
 
Property / Symbols used: Q10923 / rank
 
Normal rank
Property / Symbols used: Q10923 / qualifier
 
Defining formula:

ℂ {\displaystyle{\displaystyle\mathbb{C}}}

\Complexes
Property / Symbols used: Q10923 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r1.m2adec
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.m1adec
Property / Symbols used
 
Property / Symbols used: Q10784 / rank
 
Normal rank
Property / Symbols used: Q10784 / qualifier
 
Defining formula:

∈ {\displaystyle{\displaystyle\in}}

\in
Property / Symbols used: Q10784 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r10.m2adec
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.m2adec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2adec
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.m3adec
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.m1adec

Latest revision as of 01:51, 2 January 2020

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DLMF:7.14.E1
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e 2 ⁢ i ⁢ a ⁢ t ⁢ erfc ⁡ ( b ⁢ t ) ⁢ d t = 1 a ⁢ π ⁢ F ⁡ ( a b ) + i 2 ⁢ a ⁢ ( 1 - e - ( a / b ) 2 ) , superscript subscript 0 superscript 𝑒 2 𝑖 𝑎 𝑡 complementary-error-function 𝑏 𝑡 𝑡 1 𝑎 𝜋 Dawsons-integral 𝑎 𝑏 𝑖 2 𝑎 1 superscript 𝑒 superscript 𝑎 𝑏 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{2iat}\operatorname{erfc}\left(% bt\right)\mathrm{d}t={\frac{1}{a\sqrt{\pi}}F\left(\frac{a}{b}\right)+\frac{i}{% 2a}\left(1-e^{-(a/b)^{2}}\right)},}}
    0 references
    DLMF id
    DLMF:7.14.E1
    0 references
    constraint
    a ∈ ℂ 𝑎 {\displaystyle{\displaystyle a\in\mathbb{C}}}
    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
    Dawson’s integral
    Defining formula
    F ⁡ ( z ) Dawsons-integral 𝑧 {\displaystyle{\displaystyle F\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E5.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    complementary error function
    Defining formula
    erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}
    xml-id
    C7.S2.E2.m2adec
    0 references
    Q10923
    Defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
    0 references
     
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