Confluent Hypergeometric Functions - 13.29 Methods of Computation

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
13.29.E1 z 2 ⁢ ( n + μ - 1 2 ) ⁢ ( ( n + μ + 1 2 ) 2 - κ 2 ) ( n + μ ) ⁢ ( n + μ + 1 2 ) ⁢ ( n + μ + 1 ) ⁢ y ⁢ ( n + 1 ) + 16 ⁢ ( ( n + μ ) 2 - 1 2 ⁢ κ ⁢ z - 1 4 ) ⁢ y ⁢ ( n ) - 16 ⁢ ( ( n + μ ) 2 - 1 4 ) ⁢ y ⁢ ( n - 1 ) = 0 superscript 𝑧 2 𝑛 𝜇 1 2 superscript 𝑛 𝜇 1 2 2 superscript 𝜅 2 𝑛 𝜇 𝑛 𝜇 1 2 𝑛 𝜇 1 𝑦 𝑛 1 16 superscript 𝑛 𝜇 2 1 2 𝜅 𝑧 1 4 𝑦 𝑛 16 superscript 𝑛 𝜇 2 1 4 𝑦 𝑛 1 0 {\displaystyle{\displaystyle\frac{z^{2}(n+\mu-\tfrac{1}{2})\left((n+\mu+\tfrac% {1}{2})^{2}-\kappa^{2}\right)}{(n+\mu)(n+\mu+\tfrac{1}{2})(n+\mu+1)}{y(n+1)}+1% 6\left((n+\mu)^{2}-\tfrac{1}{2}\kappa z-\tfrac{1}{4}\right)y(n)\\ -16\left((n+\mu)^{2}-\tfrac{1}{4}\right)y(n-1)=0}}
\frac{z^{2}(n+\mu-\tfrac{1}{2})\left((n+\mu+\tfrac{1}{2})^{2}-\kappa^{2}\right)}{(n+\mu)(n+\mu+\tfrac{1}{2})(n+\mu+1)}{y(n+1)}+16\left((n+\mu)^{2}-\tfrac{1}{2}\kappa z-\tfrac{1}{4}\right)y(n)\\ -16\left((n+\mu)^{2}-\tfrac{1}{4}\right)y(n-1) = 0		 

((x + y*I)^(2)*(n + mu -(1)/(2))*((n + mu +(1)/(2))^(2)- (kappa)^(2)))/((n + mu)*(n + mu +(1)/(2))*(n + mu + 1))*y*(n + 1)+ 16*((n + mu)^(2)-(1)/(2)*kappa*(x + y*I)-(1)/(4))*((x + y*I)^(- n - mu -(1)/(2))* WhittakerM(kappa, n + mu, x + y*I))*; - 16*((n + mu)^(2)-(1)/(4))*y*(n - 1) = 0
 
Divide[(x + y*I)^(2)*(n + \[Mu]-Divide[1,2])*((n + \[Mu]+Divide[1,2])^(2)- \[Kappa]^(2)),(n + \[Mu])*(n + \[Mu]+Divide[1,2])*(n + \[Mu]+ 1)]*y*(n + 1)+ 16*((n + \[Mu])^(2)-Divide[1,2]*\[Kappa]*(x + y*I)-Divide[1,4])*((x + y*I)^(- n - \[Mu]-Divide[1,2])* WhittakerM[\[Kappa], n + \[Mu], x + y*I])*- 16*((n + \[Mu])^(2)-Divide[1,4])*y*(n - 1) == 0
 Skipped - no semantic math Skipped - no semantic math - -
13.29.E3 e - 1 2 ⁢ z = ∑ s = 0 ∞ ( 2 ⁢ μ ) s ⁢ ( 1 2 + μ - κ ) s ( 2 ⁢ μ ) 2 ⁢ s ⁢ s ! ⁢ ( - z ) s ⁢ y ⁢ ( s ) superscript 𝑒 1 2 𝑧 superscript subscript 𝑠 0 Pochhammer 2 𝜇 𝑠 Pochhammer 1 2 𝜇 𝜅 𝑠 Pochhammer 2 𝜇 2 𝑠 𝑠 superscript 𝑧 𝑠 𝑦 𝑠 {\displaystyle{\displaystyle e^{-\frac{1}{2}z}=\sum_{s=0}^{\infty}\frac{{\left% (2\mu\right)_{s}}{\left(\frac{1}{2}+\mu-\kappa\right)_{s}}}{{\left(2\mu\right)% _{2s}}s!}(-z)^{s}y(s)}}
e^{-\frac{1}{2}z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{2\mu}{s}\Pochhammersym{\frac{1}{2}+\mu-\kappa}{s}}{\Pochhammersym{2\mu}{2s}s!}(-z)^{s}y(s)		 

exp(-(1)/(2)*(x + y(I))) = sum((pochhammer(2*mu, s)*pochhammer((1)/(2)+ mu - kappa, s))/(pochhammer(2*mu, 2*s)*factorial(s))*(-(x + y(I)))^(s)* y(s), s = 0..infinity)

Exp[-Divide[1,2]*(x + y[I])] == Sum[Divide[Pochhammer[2*\[Mu], s]*Pochhammer[Divide[1,2]+ \[Mu]- \[Kappa], s],Pochhammer[2*\[Mu], 2*s]*(s)!]*(-(x + y[I]))^(s)* y[s], {s, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: .505394540e-1+.5994002652*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}


Result: .7100232023-.2722368431*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.0505394539002913, 0.5994002653939074]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.9437946777348876, -0.07485124664222054]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
13.29.E5 ( n + a ) ⁢ w ⁢ ( n ) - ( 2 ⁢ ( n + a + 1 ) + z - b ) ⁢ w ⁢ ( n + 1 ) + ( n + a - b + 2 ) ⁢ w ⁢ ( n + 2 ) = 0 𝑛 𝑎 𝑤 𝑛 2 𝑛 𝑎 1 𝑧 𝑏 𝑤 𝑛 1 𝑛 𝑎 𝑏 2 𝑤 𝑛 2 0 {\displaystyle{\displaystyle(n+a)w(n)-\left(2(n+a+1)+z-b\right)w(n+1)+(n+a-b+2% )w(n+2)=0}}
(n+a)w(n)-\left(2(n+a+1)+z-b\right)w(n+1)+(n+a-b+2)w(n+2) = 0		 

(n + a)*w(n)-(2*(n + a + 1)+ z - b)*w(n + 1)+(n + a - b + 2)*w(n + 2) = 0
 
(n + a)*w[n]-(2*(n + a + 1)+ z - b)*w[n + 1]+(n + a - b + 2)*w[n + 2] == 0
 Skipped - no semantic math Skipped - no semantic math - -
13.29.E6 w ⁢ ( n ) = ( a ) n ⁢ U ⁡ ( n + a , b , z ) 𝑤 𝑛 Pochhammer 𝑎 𝑛 Kummer-confluent-hypergeometric-U 𝑛 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle w(n)={\left(a\right)_{n}}U\left(n+a,b,z\right)}}
w(n) = \Pochhammersym{a}{n}\KummerconfhyperU@{n+a}{b}{z}		 

w(n) = pochhammer(a, n)*KummerU(n + a, b, z)

w[n] == Pochhammer[a, n]*HypergeometricU[n + a, b, z]
Failure Failure
Failed [300 / 300]
Result: 3.350777422+.7382256467*I
Test Values: {a = -3/2, b = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 1.327538097+1.034245119*I
Test Values: {a = -3/2, b = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[3.3507774204902745, 0.7382256467588033]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.3275380963595516, 1.0342451193960447]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
13.29.E7 z - a = ∑ s = 0 ∞ ( a - b + 1 ) s s ! ⁢ w ⁢ ( s ) superscript 𝑧 𝑎 superscript subscript 𝑠 0 Pochhammer 𝑎 𝑏 1 𝑠 𝑠 𝑤 𝑠 {\displaystyle{\displaystyle z^{-a}=\sum_{s=0}^{\infty}\frac{{\left(a-b+1% \right)_{s}}}{s!}w(s)}}
z^{-a} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a-b+1}{s}}{s!}w(s)		 

(z)^(- a) = sum((pochhammer(a - b + 1, s))/(factorial(s))*w(s), s = 0..infinity)

(z)^(- a) == Sum[Divide[Pochhammer[a - b + 1, s],(s)!]*w[s], {s, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -3/2, b = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -3/2, b = -3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

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


Result: DirectedInfinity[]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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