DLMF:19.9.E11 (Q6254): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: inverse Gudermannian function / rank
 
Normal rank
Property / Symbols used: inverse Gudermannian function / qualifier
 
test:

gd - 1 ( x ) inverse-Gudermannian 𝑥 {\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(\NVar{x}\right)}}

\aGudermannian@{\NVar{x}}
Property / Symbols used: inverse Gudermannian function / qualifier
 
xml-id: C4.S23.E41.m2adec
Property / Symbols used
 
Property / Symbols used: Q11807 / rank
 
Normal rank
Property / Symbols used: Q11807 / qualifier
 
test:

ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}

\phi
Property / Symbols used: Q11807 / qualifier
 
xml-id: C19.S1.XMD4.m1dec
Property / Symbols used
 
Property / Symbols used: Q11808 / rank
 
Normal rank
Property / Symbols used: Q11808 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11808 / qualifier
 
xml-id: C19.S1.XMD5.m1idec
Property / Symbols used
 
Property / Symbols used: Q11839 / rank
 
Normal rank
Property / Symbols used: Q11839 / qualifier
 
test:

Δ Δ {\displaystyle{\displaystyle\Delta}}

\Delta
Property / Symbols used: Q11839 / qualifier
 
xml-id: C19.S9.XMD2.m1dec

Latest revision as of 12:45, 2 January 2020

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DLMF:19.9.E11
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    Statements

    ϕ F ( ϕ , k ) min ( ϕ / Δ , gd - 1 ( ϕ ) ) , italic-ϕ elliptic-integral-first-kind-F italic-ϕ 𝑘 italic-ϕ Δ inverse-Gudermannian italic-ϕ {\displaystyle{\displaystyle\phi\leq F\left(\phi,k\right)\leq\min(\phi/\Delta,% {\operatorname{gd}^{-1}}\left(\phi\right)),}}
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    F ( ϕ , k ) elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}
    C19.S2.E4.m2adec
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    gd - 1 ( x ) inverse-Gudermannian 𝑥 {\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(\NVar{x}\right)}}
    C4.S23.E41.m2adec
    0 references
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    C19.S1.XMD4.m1dec
    0 references
    k 𝑘 {\displaystyle{\displaystyle k}}
    C19.S1.XMD5.m1idec
    0 references
    Δ Δ {\displaystyle{\displaystyle\Delta}}
    C19.S9.XMD2.m1dec
    0 references