Confluent Hypergeometric Functions - 13.8 Asymptotic Approximations for Large Parameters

From testwiki
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
13.8.E3 ( e t - 1 ) a - 1 ⁢ exp ⁡ ( t + z ⁢ ( 1 - e - t ) ) = ∑ s = 0 ∞ q s ⁢ ( z , a ) ⁢ t s + a - 1 superscript superscript 𝑒 𝑡 1 𝑎 1 𝑡 𝑧 1 superscript 𝑒 𝑡 superscript subscript 𝑠 0 subscript 𝑞 𝑠 𝑧 𝑎 superscript 𝑡 𝑠 𝑎 1 {\displaystyle{\displaystyle\left(e^{t}-1\right)^{a-1}\exp\left(t+z(1-e^{-t})% \right)=\sum_{s=0}^{\infty}q_{s}(z,a)t^{s+a-1}}}
\left(e^{t}-1\right)^{a-1}\exp@{t+z(1-e^{-t})} = \sum_{s=0}^{\infty}q_{s}(z,a)t^{s+a-1}		 

(exp(t)- 1)^(a - 1)* exp(t + z*(1 - exp(- t))) = sum(q[s](z , a)* (t)^(s + a - 1), s = 0..infinity)

(Exp[t]- 1)^(a - 1)* Exp[t + z*(1 - Exp[- t])] == Sum[Subscript[q, s][z , a]* (t)^(s + a - 1), {s, 0, Infinity}, GenerateConditions->None]
Error Failure - Error
13.8#Ex1 p k ⁢ ( z ) = ∑ s = 0 k ( k s ) ⁢ ( 1 - b + s ) k - s ⁢ z s ⁢ c k + s ⁢ ( z ) subscript 𝑝 𝑘 𝑧 superscript subscript 𝑠 0 𝑘 binomial 𝑘 𝑠 Pochhammer 1 𝑏 𝑠 𝑘 𝑠 superscript 𝑧 𝑠 subscript 𝑐 𝑘 𝑠 𝑧 {\displaystyle{\displaystyle p_{k}(z)=\sum_{s=0}^{k}\genfrac{(}{)}{0.0pt}{}{k}% {s}{\left(1-b+s\right)_{k-s}}z^{s}c_{k+s}(z)}}
p_{k}(z) = \sum_{s=0}^{k}\binom{k}{s}\Pochhammersym{1-b+s}{k-s}z^{s}c_{k+s}(z)		 

p[k](z) = sum(binomial(k,s)*pochhammer(1 - b + s, k - s)*(z)^(s)* c[k + s](z), s = 0..k)

Subscript[p, k][z] == Sum[Binomial[k,s]*Pochhammer[1 - b + s, k - s]*(z)^(s)* Subscript[c, k + s][z], {s, 0, k}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: -.7500000009-2.299038107*I
Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[k+s] = 1/2*3^(1/2)+1/2*I, p[k] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -3.375000005-14.57772229*I
Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[k+s] = 1/2*3^(1/2)+1/2*I, p[k] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
 Skipped - Because timed out
13.8#Ex2 q k ⁢ ( z ) = ∑ s = 0 k ( k s ) ⁢ ( 2 - b + s ) k - s ⁢ z s ⁢ c k + s + 1 ⁢ ( z ) subscript 𝑞 𝑘 𝑧 superscript subscript 𝑠 0 𝑘 binomial 𝑘 𝑠 Pochhammer 2 𝑏 𝑠 𝑘 𝑠 superscript 𝑧 𝑠 subscript 𝑐 𝑘 𝑠 1 𝑧 {\displaystyle{\displaystyle q_{k}(z)=\sum_{s=0}^{k}\genfrac{(}{)}{0.0pt}{}{k}% {s}{\left(2-b+s\right)_{k-s}}z^{s}c_{k+s+1}(z)}}
q_{k}(z) = \sum_{s=0}^{k}\binom{k}{s}\Pochhammersym{2-b+s}{k-s}z^{s}c_{k+s+1}(z)		 

q[k](z) = sum(binomial(k,s)*pochhammer(2 - b + s, k - s)*(z)^(s)* c[k + s + 1](z), s = 0..k)

Subscript[q, k][z] == Sum[Binomial[k,s]*Pochhammer[2 - b + s, k - s]*(z)^(s)* Subscript[c, k + s + 1][z], {s, 0, k}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: -1.250000001-3.165063511*I
Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[k+s+1] = 1/2*3^(1/2)+1/2*I, q[k] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -6.875000009-22.63990012*I
Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[k+s+1] = 1/2*3^(1/2)+1/2*I, q[k] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
 Skipped - Because timed out
13.8.E16 ( k + 1 ) ⁢ c k + 1 ⁢ ( z ) + ∑ s = 0 k ( b ⁢ B s + 1 ( s + 1 ) ! + z ⁢ ( s + 1 ) ⁢ B s + 2 ( s + 2 ) ! ) ⁢ c k - s ⁢ ( z ) = 0 𝑘 1 subscript 𝑐 𝑘 1 𝑧 superscript subscript 𝑠 0 𝑘 𝑏 Bernoulli-number-B 𝑠 1 𝑠 1 𝑧 𝑠 1 Bernoulli-number-B 𝑠 2 𝑠 2 subscript 𝑐 𝑘 𝑠 𝑧 0 {\displaystyle{\displaystyle(k+1)c_{k+1}(z)+\sum_{s=0}^{k}\left(\frac{bB_{s+1}% }{(s+1)!}+\frac{z(s+1)B_{s+2}}{(s+2)!}\right)c_{k-s}(z)=0}}
(k+1)c_{k+1}(z)+\sum_{s=0}^{k}\left(\frac{b\BernoullinumberB{s+1}}{(s+1)!}+\frac{z(s+1)\BernoullinumberB{s+2}}{(s+2)!}\right)c_{k-s}(z) = 0		 

(k + 1)*c[k + 1](z)+ sum(((b*bernoulli(s + 1))/(factorial(s + 1))+(z*(s + 1)*bernoulli(s + 2))/(factorial(s + 2)))*c[k - s](z), s = 0..k) = 0

(k + 1)*Subscript[c, k + 1][z]+ Sum[(Divide[b*BernoulliB[s + 1],(s + 1)!]+Divide[z*(s + 1)*BernoulliB[s + 2],(s + 2)!])*Subscript[c, k - s][z], {s, 0, k}, GenerateConditions->None] == 0
Failure Failure
Failed [300 / 300]
Result: 2.313541668+4.086338379*I
Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[1+k] = 1/2*3^(1/2)+1/2*I, c[k-s] = 1/2*3^(1/2)+1/2*I, k = 3}


Result: 1.377763239+3.777643283*I
Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[1+k] = 1/2*3^(1/2)+1/2*I, c[k-s] = -1/2+1/2*I*3^(1/2), k = 3}

... skip entries to safe data
 Skipped - Because timed out
13.8#Ex3 ∂ ⁡ f ∂ ⁡ t = ( b ⁢ ( 1 t - 1 e t - 1 ) - z ⁢ ( 1 t 2 - e t ( e t - 1 ) 2 ) ) ⁢ f partial-derivative 𝑓 𝑡 𝑏 1 𝑡 1 superscript 𝑒 𝑡 1 𝑧 1 superscript 𝑡 2 superscript 𝑒 𝑡 superscript superscript 𝑒 𝑡 1 2 𝑓 {\displaystyle{\displaystyle\frac{\partial f}{\partial t}=\left(b\left(\frac{1% }{t}-\frac{1}{e^{t}-1}\right)-z\left(\frac{1}{t^{2}}-\frac{e^{t}}{\left(e^{t}-% 1\right)^{2}}\right)\right)f}}
\pderiv{f}{t} = \left(b\left(\frac{1}{t}-\frac{1}{e^{t}-1}\right)-z\left(\frac{1}{t^{2}}-\frac{e^{t}}{\left(e^{t}-1\right)^{2}}\right)\right)f		 

diff(f, t) = (b*((1)/(t)-(1)/(exp(t)- 1))- z*((1)/((t)^(2))-(exp(t))/((exp(t)- 1)^(2))))*f

D[f, t] == (b*(Divide[1,t]-Divide[1,Exp[t]- 1])- z*(Divide[1,(t)^(2)]-Divide[Exp[t],(Exp[t]- 1)^(2)]))*f
Failure Failure
Failed [300 / 300]
Result: .8434854075+.5301342049*I
Test Values: {b = -3/2, f = 1/2*3^(1/2)+1/2*I, t = -3/2, z = 1/2*3^(1/2)+1/2*I}


Result: .7413969054+.5027796732*I
Test Values: {b = -3/2, f = 1/2*3^(1/2)+1/2*I, t = -3/2, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.8434854065788572, 0.5301342044541701]
Test Values: {Rule[b, -1.5], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.7413969045334019, 0.5027796727745873]
Test Values: {Rule[b, -1.5], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
Retrieved from "https://lct.wmflabs.org/w/index.php?title=13.8&oldid=58257"