DLMF:10.27.E11 (Q3501): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(4 intermediate revisions by the same user not shown)
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.m2afdec
Property / Symbols used
 
Property / Symbols used: modified Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
Defining formula:

K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modBesselK{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
xml-id: C10.S25.E3.m2agdec
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.m1aedec
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.m1jdec
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.m1idec

Latest revision as of 13:37, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:10.27.E11
No description defined

    Statements

    Defining formula
    Y ν ⁡ ( z ) = e ± ( ν + 1 ) ⁢ π ⁢ i / 2 ⁢ I ν ⁡ ( z ⁢ e ∓ π ⁢ i / 2 ) - ( 2 / π ) ⁢ e ∓ ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( z ⁢ e ∓ π ⁢ i / 2 ) , Bessel-Y-Weber 𝜈 𝑧 superscript 𝑒 plus-or-minus 𝜈 1 𝜋 𝑖 2 modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 minus-or-plus 𝜋 𝑖 2 2 𝜋 superscript 𝑒 minus-or-plus 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 minus-or-plus 𝜋 𝑖 2 {\displaystyle{\displaystyle Y_{\nu}\left(z\right)=e^{\pm(\nu+1)\pi i/2}I_{\nu% }\left(ze^{\mp\pi i/2}\right)-(2/\pi)e^{\mp\nu\pi i/2}K_{\nu}\left(ze^{\mp\pi i% /2}\right),}}
    0 references
    DLMF id
    DLMF:10.27.E11
    0 references
    constraint
    - 1 2 ⁢ π ≤ ± ph ⁡ z ≤ π 1 2 𝜋 plus-or-minus phase 𝑧 𝜋 {\displaystyle{\displaystyle-\tfrac{1}{2}\pi\leq\pm\operatorname{ph}z\leq\pi}}
    0 references
    Symbols used
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    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.m2afdec
    0 references
    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2agdec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aedec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1jdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1idec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q3501&oldid=38318"