DLMF:7.14.E4 (Q2436): Difference between revisions

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Property / constraint
 

ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}

\realpart@@{b}>0
Property / constraint: ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}} / rank
 
Normal rank
Property / constraint
 

ℜ ⁡ c ≥ 0 𝑐 0 {\displaystyle{\displaystyle\Re c\geq 0}}

\realpart@@{c}\geq 0
Property / constraint: ℜ ⁡ c ≥ 0 𝑐 0 {\displaystyle{\displaystyle\Re c\geq 0}} / rank
 
Normal rank
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: 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.m1acdec
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.m2acdec
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.m3acdec
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.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

Latest revision as of 01:51, 2 January 2020

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DLMF:7.14.E4
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e ( a - b ) ⁢ t ⁢ erfc ⁡ ( a ⁢ t + c t ) ⁢ d t = e - 2 ⁢ ( a ⁢ c + b ⁢ c ) b ⁢ ( a + b ) , superscript subscript 0 superscript 𝑒 𝑎 𝑏 𝑡 complementary-error-function 𝑎 𝑡 𝑐 𝑡 𝑡 superscript 𝑒 2 𝑎 𝑐 𝑏 𝑐 𝑏 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{(a-b)t}\operatorname{erfc}% \left(\sqrt{at}+\sqrt{\frac{c}{t}}\right)\mathrm{d}t=\frac{e^{-2(\sqrt{ac}+% \sqrt{bc})}}{\sqrt{b}(\sqrt{a}+\sqrt{b})},}}
    0 references
    DLMF id
    DLMF:7.14.E4
    0 references
    constraint
    | ph ⁡ a | < 1 2 ⁢ π phase 𝑎 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}a|<\frac{1}{2}\pi}}
    0 references
    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
    0 references
    ℜ ⁡ c ≥ 0 𝑐 0 {\displaystyle{\displaystyle\Re c\geq 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.m2abdec
    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.m1acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1abdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
    0 references
     
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