DLMF:7.7.E15 (Q2385): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
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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.m2aldec
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.m3andec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1addec

Latest revision as of 01:46, 2 January 2020

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DLMF:7.7.E15
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    Statements

    0 e - a t cos ( t 2 ) d t = π 2 f ( a 2 π ) , superscript subscript 0 superscript 𝑒 𝑎 𝑡 superscript 𝑡 2 𝑡 𝜋 2 Fresnel-auxilliary-function-f 𝑎 2 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\cos\left(t^{2}\right)% \mathrm{d}t=\sqrt{\frac{\pi}{2}}\mathrm{f}\left(\frac{a}{\sqrt{2\pi}}\right),}}
    0 references
    DLMF:7.7.E15
    0 references
    a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    0 references
    f ( z ) Fresnel-auxilliary-function-f 𝑧 {\displaystyle{\displaystyle\mathrm{f}\left(\NVar{z}\right)}}
    C7.S2.E10.m2acdec
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    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2andec
    0 references
    cos z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    C4.S14.E2.m2adec
    0 references
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    C1.S4.SS4.m1andec
    0 references
    e {\displaystyle{\displaystyle\mathrm{e}}}
    C4.S2.E11.m2aldec
    0 references
    {\displaystyle{\displaystyle\int}}
    C1.S4.SS4.m3andec
    0 references
    absent {\displaystyle{\displaystyle\Re}}
    C1.S9.E2.m1addec
    0 references