DLMF:7.5.E12 (Q2358): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: auxiliary function for Fresnel integrals / rank
 
Normal rank
Property / Symbols used: auxiliary function for Fresnel integrals / qualifier
 
test:

f ⁡ ( z ) Fresnel-auxilliary-function-f 𝑧 {\displaystyle{\displaystyle\mathrm{f}\left(\NVar{z}\right)}}

\auxFresnelf@{\NVar{z}}
Property / Symbols used: auxiliary function for Fresnel integrals / qualifier
 
xml-id: C7.S2.E10.m2afdec
Property / Symbols used
 
Property / Symbols used: auxiliary function for Fresnel integrals / rank
 
Normal rank
Property / Symbols used: auxiliary function for Fresnel integrals / qualifier
 
test:

g ⁡ ( z ) Fresnel-auxilliary-function-g 𝑧 {\displaystyle{\displaystyle\mathrm{g}\left(\NVar{z}\right)}}

\auxFresnelg@{\NVar{z}}
Property / Symbols used: auxiliary function for Fresnel integrals / qualifier
 
xml-id: C7.S2.E11.m2afdec
Property / Symbols used
 
Property / Symbols used: Fresnel integral / rank
 
Normal rank
Property / Symbols used: Fresnel integral / qualifier
 
test:

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

\FresnelintF@{\NVar{z}}
Property / Symbols used: Fresnel integral / qualifier
 
xml-id: C7.S2.E6.m2acdec
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
 
test:

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.m2agdec
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
test:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11313 / rank
 
Normal rank
Property / Symbols used: Q11313 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11313 / qualifier
 
xml-id: C7.S1.XMD1.m1adec

Latest revision as of 00:40, 2 January 2020

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DLMF:7.5.E12
No description defined

    Statements

    test
    | ℱ ⁡ ( x ) | 2 = 2 + f 2 ⁡ ( - x ) + g 2 ⁡ ( - x ) - 2 ⁢ 2 ⁢ cos ⁡ ( 1 4 ⁢ π + 1 2 ⁢ π ⁢ x 2 ) ⁢ f ⁡ ( - x ) - 2 ⁢ 2 ⁢ cos ⁡ ( 1 4 ⁢ π - 1 2 ⁢ π ⁢ x 2 ) ⁢ g ⁡ ( - x ) , superscript Fresnel-integral 𝑥 2 2 Fresnel-auxilliary-function-f 2 𝑥 Fresnel-auxilliary-function-g 2 𝑥 2 2 1 4 𝜋 1 2 𝜋 superscript 𝑥 2 Fresnel-auxilliary-function-f 𝑥 2 2 1 4 𝜋 1 2 𝜋 superscript 𝑥 2 Fresnel-auxilliary-function-g 𝑥 {\displaystyle{\displaystyle|\mathcal{F}\left(x\right)|^{2}=2+{\mathrm{f}^{2}}% \left(-x\right)+{\mathrm{g}^{2}}\left(-x\right)-2\sqrt{2}\cos\left(\tfrac{1}{4% }\pi+\tfrac{1}{2}\pi x^{2}\right)\mathrm{f}\left(-x\right)-2\sqrt{2}\cos\left(% \tfrac{1}{4}\pi-\tfrac{1}{2}\pi x^{2}\right)\mathrm{g}\left(-x\right),}}
    0 references
    DLMF id
    DLMF:7.5.E12
    0 references
    constraint
    x ≤ 0 𝑥 0 {\displaystyle{\displaystyle x\leq 0}}
    0 references
    Symbols used
    auxiliary function for Fresnel integrals
    test
    f ⁡ ( z ) Fresnel-auxilliary-function-f 𝑧 {\displaystyle{\displaystyle\mathrm{f}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E10.m2afdec
    0 references
    auxiliary function for Fresnel integrals
    test
    g ⁡ ( z ) Fresnel-auxilliary-function-g 𝑧 {\displaystyle{\displaystyle\mathrm{g}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E11.m2afdec
    0 references
    Fresnel integral
    test
    ℱ ⁡ ( z ) Fresnel-integral 𝑧 {\displaystyle{\displaystyle\mathcal{F}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E6.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    Q11313
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C7.S1.XMD1.m1adec
    0 references
     
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