DLMF:7.18.E4 (Q2454): Difference between revisions

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Property / constraint
 

n = 0 , 1 , 2 , 𝑛 0 1 2 {\displaystyle{\displaystyle n=0,1,2,\dots}}

n=0,1,2,\dots
Property / constraint: n = 0 , 1 , 2 , 𝑛 0 1 2 {\displaystyle{\displaystyle n=0,1,2,\dots}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: complementary error function / rank
 
Normal rank
Property / Symbols used: complementary error function / qualifier
 
test:

erfc z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}

\erfc@@{\NVar{z}}
Property / Symbols used: complementary error function / qualifier
 
xml-id: C7.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / rank
 
Normal rank
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
test:

d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}

\deriv{\NVar{f}}{\NVar{x}}
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
xml-id: C1.S4.E4.m2aadec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5aadec
Property / Symbols used
 
Property / Symbols used: repeated integrals of the complementary error function / rank
 
Normal rank
Property / Symbols used: repeated integrals of the complementary error function / qualifier
 
test:

i n erfc ( z ) repeated-integral-complementary-error-function 𝑛 𝑧 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(% \NVar{z}\right)}}

\repinterfc{\NVar{n}}@{\NVar{z}}
Property / Symbols used: repeated integrals of the complementary error function / qualifier
 
xml-id: C7.S18.E2.m2addec
Property / Symbols used
 
Property / Symbols used: Q11312 / rank
 
Normal rank
Property / Symbols used: Q11312 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11312 / qualifier
 
xml-id: C7.S1.XMD2.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11314 / rank
 
Normal rank
Property / Symbols used: Q11314 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11314 / qualifier
 
xml-id: C7.S1.XMD3.m1cdec

Latest revision as of 12:31, 2 January 2020

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DLMF:7.18.E4
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    Statements

    d n d z n ( e z 2 erfc z ) = ( - 1 ) n 2 n n ! e z 2 i n erfc ( z ) , derivative 𝑧 𝑛 superscript 𝑒 superscript 𝑧 2 complementary-error-function 𝑧 superscript 1 𝑛 superscript 2 𝑛 𝑛 superscript 𝑒 superscript 𝑧 2 repeated-integral-complementary-error-function 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {z^{2}}\operatorname{erfc}z\right)=(-1)^{n}2^{n}n!e^{z^{2}}\mathop{\mathrm{i}^% {n}\mathrm{erfc}}\left(z\right),}}
    0 references
    0 references
    n = 0 , 1 , 2 , 𝑛 0 1 2 {\displaystyle{\displaystyle n=0,1,2,\dots}}
    0 references
    erfc z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}
    C7.S2.E2.m2aadec
    0 references
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    C1.S4.E4.m2aadec
    0 references
    e {\displaystyle{\displaystyle\mathrm{e}}}
    C4.S2.E11.m2abdec
    0 references
    ! {\displaystyle{\displaystyle!}}
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    i n erfc ( z ) repeated-integral-complementary-error-function 𝑛 𝑧 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(% \NVar{z}\right)}}
    C7.S18.E2.m2addec
    0 references
    z 𝑧 {\displaystyle{\displaystyle z}}
    C7.S1.XMD2.m1cdec
    0 references
    n 𝑛 {\displaystyle{\displaystyle n}}
    C7.S1.XMD3.m1cdec
    0 references