DLMF:7.18.E14 (Q2464): Difference between revisions

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Property / constraint
 

| ph ⁡ z | ≤ 3 4 ⁢ π - δ ( < 3 4 ⁢ π ) phase 𝑧 annotated 3 4 𝜋 𝛿 absent 3 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<% \tfrac{3}{4}\pi)}}

|\phase@@{z}|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)
Property / constraint: | ph ⁡ z | ≤ 3 4 ⁢ π - δ ( < 3 4 ⁢ π ) phase 𝑧 annotated 3 4 𝜋 𝛿 absent 3 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<% \tfrac{3}{4}\pi)}} / rank
 
Normal rank
Property / constraint
 

z → ∞ → 𝑧 {\displaystyle{\displaystyle z\to\infty}}

z\to\infty
Property / constraint: z → ∞ → 𝑧 {\displaystyle{\displaystyle z\to\infty}} / rank
 
Normal rank
Property / constraint
 

| ph ⁡ z | ≤ 3 4 ⁢ π - δ ( < 3 4 ⁢ π ) ph 𝑧 annotated 3 4 𝜋 𝛿 absent 3 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<% \tfrac{3}{4}\pi)}}

|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)
Property / constraint: | ph ⁡ z | ≤ 3 4 ⁢ π - δ ( < 3 4 ⁢ π ) ph 𝑧 annotated 3 4 𝜋 𝛿 absent 3 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<% \tfrac{3}{4}\pi)}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q10931 / rank
 
Normal rank
Property / Symbols used: Q10931 / qualifier
 
test:

∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}

\asympexp
Property / Symbols used: Q10931 / qualifier
 
xml-id: C2.S1.SS3.p1.m11adec
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.m2aedec
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.m2afdec
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.m5addec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
test:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1adec
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.m2andec
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.m1mdec
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.m1mdec

Latest revision as of 12:33, 2 January 2020

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DLMF:7.18.E14
No description defined

    Statements

    test
    i n ⁢ erfc ⁡ ( z ) ∼ 2 π ⁢ e - z 2 ( 2 ⁢ z ) n + 1 ⁢ ∑ m = 0 ∞ ( - 1 ) m ⁢ ( 2 ⁢ m + n ) ! n ! ⁢ m ! ⁢ ( 2 ⁢ z ) 2 ⁢ m , asymptotic-expansion repeated-integral-complementary-error-function 𝑛 𝑧 2 𝜋 superscript 𝑒 superscript 𝑧 2 superscript 2 𝑧 𝑛 1 superscript subscript 𝑚 0 superscript 1 𝑚 2 𝑚 𝑛 𝑛 𝑚 superscript 2 𝑧 2 𝑚 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)% \sim\frac{2}{\sqrt{\pi}}\frac{e^{-z^{2}}}{(2z)^{n+1}}\sum_{m=0}^{\infty}\frac{% (-1)^{m}(2m+n)!}{n!m!(2z)^{2m}},}}
    0 references
    DLMF id
    DLMF:7.18.E14
    0 references
    constraint
    | ph ⁡ z | ≤ 3 4 ⁢ π - δ ( < 3 4 ⁢ π ) phase 𝑧 annotated 3 4 𝜋 𝛿 absent 3 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<% \tfrac{3}{4}\pi)}}
    0 references
    z → ∞ → 𝑧 {\displaystyle{\displaystyle z\to\infty}}
    0 references
    | ph ⁡ z | ≤ 3 4 ⁢ π - δ ( < 3 4 ⁢ π ) ph 𝑧 annotated 3 4 𝜋 𝛿 absent 3 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<% \tfrac{3}{4}\pi)}}
    0 references
    Symbols used
    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
    0 references
    repeated integrals of the complementary error function
    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)}}
    xml-id
    C7.S18.E2.m2andec
    0 references
    Q11312
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1mdec
    0 references
    Q11314
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C7.S1.XMD3.m1mdec
    0 references
     
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