DLMF:13.2.E41 (Q4333): Difference between revisions

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‎Changed an Item: Add constraint
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Property / Symbols used
 
Property / Symbols used: Kummer confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Kummer confluent hypergeometric function / qualifier
 
Defining formula:

U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}

\KummerconfhyperU@{\NVar{a}}{\NVar{b}}{\NVar{z}}
Property / Symbols used: Kummer confluent hypergeometric function / qualifier
 
xml-id: C13.S2.E6.m2axdec

Revision as of 15:41, 2 January 2020

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DLMF:13.2.E41
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    Statements

    Defining formula
    1 Γ ⁡ ( b ) ⁢ M ⁡ ( a , b , z ) = e ∓ a ⁢ π ⁢ i Γ ⁡ ( b - a ) ⁢ U ⁡ ( a , b , z ) + e ± ( b - a ) ⁢ π ⁢ i Γ ⁡ ( a ) ⁢ e z ⁢ U ⁡ ( b - a , b , e ± π ⁢ i ⁢ z ) . 1 Euler-Gamma 𝑏 Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 superscript 𝑒 minus-or-plus 𝑎 𝜋 imaginary-unit Euler-Gamma 𝑏 𝑎 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑒 plus-or-minus 𝑏 𝑎 𝜋 imaginary-unit Euler-Gamma 𝑎 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 𝑏 𝑎 𝑏 superscript 𝑒 plus-or-minus 𝜋 imaginary-unit 𝑧 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(b\right)}M\left(a,b,z\right)=% \frac{e^{\mp a\pi\mathrm{i}}}{\Gamma\left(b-a\right)}U\left(a,b,z\right)+\frac% {e^{\pm(b-a)\pi\mathrm{i}}}{\Gamma\left(a\right)}e^{z}U\left(b-a,b,e^{\pm\pi% \mathrm{i}}z\right).}}
    0 references
    DLMF id
    DLMF:13.2.E41
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aqdec
    0 references
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ Kummer confluent hypergeometric function
    Defining formula
    M ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E2.m2aidec
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2axdec
    0 references
     
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