DLMF:7.6.E9 (Q2368): 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: error function / rank
 
Normal rank
Property / Symbols used: error function / qualifier
 
Defining formula:

erf ⁡ z error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erf}\NVar{z}}}

\erf@@{\NVar{z}}
Property / Symbols used: error function / qualifier
 
xml-id: C7.S2.E1.m2acdec
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.m2aadec
Property / Symbols used
 
Property / Symbols used: modified spherical Bessel function / rank
 
Normal rank
Property / Symbols used: modified spherical Bessel function / qualifier
 
Defining formula:

𝗂 n ( 1 ) ⁡ ( z ) spherical-Bessel-I-1 𝑛 𝑧 {\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)}}

\modsphBesseli{1}{\NVar{n}}@{\NVar{z}}
Property / Symbols used: modified spherical Bessel function / qualifier
 
xml-id: C10.S47.E7.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11312 / rank
 
Normal rank
Property / Symbols used: Q11312 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11312 / qualifier
 
xml-id: C7.S1.XMD2.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11314 / rank
 
Normal rank
Property / Symbols used: Q11314 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11314 / qualifier
 
xml-id: C7.S1.XMD3.m1hdec

Latest revision as of 01:42, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:7.6.E9
No description defined

    Statements

    Defining formula
    erf ⁡ ( a ⁢ z ) = 2 ⁢ z π ⁢ e ( 1 2 - a 2 ) ⁢ z 2 ⁢ ∑ n = 0 ∞ T 2 ⁢ n + 1 ⁡ ( a ) ⁢ 𝗂 n ( 1 ) ⁡ ( 1 2 ⁢ z 2 ) , error-function 𝑎 𝑧 2 𝑧 𝜋 superscript 𝑒 1 2 superscript 𝑎 2 superscript 𝑧 2 superscript subscript 𝑛 0 Chebyshev-polynomial-first-kind-T 2 𝑛 1 𝑎 spherical-Bessel-I-1 𝑛 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{erf}\left(az\right)=\frac{2z}{\sqrt{% \pi}}e^{(\frac{1}{2}-a^{2})z^{2}}\sum_{n=0}^{\infty}T_{2n+1}\left(a\right){% \mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}z^{2}\right),}}
    0 references
    DLMF id
    DLMF:7.6.E9
    0 references
    constraint
    - 1 ≤ a ≤ 1 1 𝑎 1 {\displaystyle{\displaystyle-1\leq a\leq 1}}
    0 references
    Symbols used
    Q11207
    Defining formula
    T n ⁡ ( x ) Chebyshev-polynomial-first-kind-T 𝑛 𝑥 {\displaystyle{\displaystyle T_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r4.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.m2agdec
    0 references
    error function
    Defining formula
    erf ⁡ z error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erf}\NVar{z}}}
    xml-id
    C7.S2.E1.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    modified spherical Bessel function
    Defining formula
    𝗂 n ( 1 ) ⁡ ( z ) spherical-Bessel-I-1 𝑛 𝑧 {\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S47.E7.m2aadec
    0 references
    Q11312
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1hdec
    0 references
    Q11314
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C7.S1.XMD3.m1hdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q2368&oldid=33874"