DLMF:10.27.E6 (Q3496): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(6 intermediate revisions by the same user not shown)
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: 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: modified Bessel function of the first kind / rank
 
Normal rank
Property / Symbols used: modified Bessel function of the first kind / qualifier
 
Defining formula:

I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modBesselI{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Bessel function of the first kind / qualifier
 
xml-id: C10.S25.E2.m2addec
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
Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1edec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1ddec

Latest revision as of 13:36, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:10.27.E6
No description defined

    Statements

    Defining formula
    I ν ⁡ ( z ) = e ∓ ν ⁢ π ⁢ i / 2 ⁢ J ν ⁡ ( z ⁢ e ± π ⁢ i / 2 ) , modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 minus-or-plus 𝜈 𝜋 𝑖 2 Bessel-J 𝜈 𝑧 superscript 𝑒 plus-or-minus 𝜋 𝑖 2 {\displaystyle{\displaystyle I_{\nu}\left(z\right)=e^{\mp\nu\pi i/2}J_{\nu}% \left(ze^{\pm\pi i/2}\right),}}
    0 references
    DLMF id
    DLMF:10.27.E6
    0 references
    constraint
    - π ≤ ± ph ⁡ z ≤ 1 2 ⁢ π 𝜋 plus-or-minus phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle-\pi\leq\pm\operatorname{ph}z\leq\tfrac{1}{2}\pi}}
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.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.m2abdec
    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
    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2addec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1edec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1ddec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q3496&oldid=38080"