Asymptotic Approximations - 2.4 Contour Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
2.4.E2 Q ⁢ ( z ) = ∫ 0 ∞ e - z ⁢ t ⁢ q ⁢ ( t ) ⁢ d t 𝑄 𝑧 superscript subscript 0 superscript 𝑒 𝑧 𝑡 𝑞 𝑡 𝑡 {\displaystyle{\displaystyle Q(z)=\int_{0}^{\infty}e^{-zt}q(t)\mathrm{d}t}}
Q(z) = \int_{0}^{\infty}e^{-zt}q(t)\diff{t}		 

Q(z) = int(exp(- z*t)*q(t), t = 0..infinity)

Q[z] == Integrate[Exp[- z*t]*q[t], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [292 / 300]
Result: -.3660254032+1.366025404*I
Test Values: {Q = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: Float(undefined)+.5000000004*I
Test Values: {Q = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: 1.732050808-1.000000001*I
Test Values: {Q = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}


Result: Float(undefined)-.8660254040*I
Test Values: {Q = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [284 / 300]
Result: Complex[-0.3660254037844386, 1.3660254037844386]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.7320508075688774, -0.9999999999999999]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
2.4.E5 q ⁢ ( t ) = 1 2 ⁢ π ⁢ i ⁢ ∫ σ - i ⁢ ∞ σ + i ⁢ ∞ e t ⁢ z ⁢ Q ⁢ ( z ) ⁢ d z 𝑞 𝑡 1 2 𝜋 𝑖 superscript subscript 𝜎 𝑖 𝜎 𝑖 superscript 𝑒 𝑡 𝑧 𝑄 𝑧 𝑧 {\displaystyle{\displaystyle q(t)=\frac{1}{2\pi i}\int_{\sigma-i\infty}^{% \sigma+i\infty}e^{tz}Q(z)\mathrm{d}z}}
q(t) = \frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}e^{tz}Q(z)\diff{z}		 0 < t , t < ∞ formulae-sequence 0 𝑡 𝑡 {\displaystyle{\displaystyle 0<t,t<\infty}} 
q(t) = (1)/(2*Pi*I)*int(exp(t*z)*Q(z), z = sigma - I*infinity..sigma + I*infinity)

q[t] == Divide[1,2*Pi*I]*Integrate[Exp[t*z]*Q[z], {z, \[Sigma]- I*Infinity, \[Sigma]+ I*Infinity}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: 1.299038106+.7500000000*I
Test Values: {Q = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, t = 1.5}


Result: .4330127020+.2500000000*I
Test Values: {Q = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, t = .5}


Result: 1.732050808+1.*I
Test Values: {Q = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, t = 2}


Result: 1.299038106+.7500000000*I
Test Values: {Q = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, sigma = -1/2+1/2*I*3^(1/2), t = 1.5}

... skip entries to safe data
 Skipped - Because timed out
2.4#Ex1 p ⁢ ( t ) = p ⁢ ( a ) + ∑ s = 0 ∞ p s ⁢ ( t - a ) s + μ 𝑝 𝑡 𝑝 𝑎 superscript subscript 𝑠 0 subscript 𝑝 𝑠 superscript 𝑡 𝑎 𝑠 𝜇 {\displaystyle{\displaystyle p(t)=p(a)+\sum_{s=0}^{\infty}p_{s}(t-a)^{s+\mu}}}
p(t) = p(a)+\sum_{s=0}^{\infty}p_{s}(t-a)^{s+\mu}		 

p(t) = p(a)+ sum((p[s](t - a))^(s + mu), s = 0..infinity)
 
p[t] == p[a]+ Sum[(Subscript[p, s][t - a])^(s + \[Mu]), {s, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
2.4#Ex2 q ⁢ ( t ) = ∑ s = 0 ∞ q s ⁢ ( t - a ) s + λ - 1 𝑞 𝑡 superscript subscript 𝑠 0 subscript 𝑞 𝑠 superscript 𝑡 𝑎 𝑠 𝜆 1 {\displaystyle{\displaystyle q(t)=\sum_{s=0}^{\infty}q_{s}(t-a)^{s+\lambda-1}}}
q(t) = \sum_{s=0}^{\infty}q_{s}(t-a)^{s+\lambda-1}		 

q(t) = sum((q[s](t - a))^(s + lambda - 1), s = 0..infinity)
 
q[t] == Sum[(Subscript[q, s][t - a])^(s + \[Lambda]- 1), {s, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
2.4.E13 | θ + μ ⁢ ω + ph ⁡ p 0 | ≤ 1 2 ⁢ π 𝜃 𝜇 𝜔 phase subscript 𝑝 0 1 2 𝜋 {\displaystyle{\displaystyle|\theta+\mu\omega+\operatorname{ph}p_{0}|\leq% \tfrac{1}{2}\pi}}
|\theta+\mu\omega+\phase@@{p_{0}}| \leq \tfrac{1}{2}\pi		 

abs(theta + mu*omega + argument(p[0])) <= (1)/(2)*Pi

Abs[\[Theta]+ \[Mu]*\[Omega]+ Arg[Subscript[p, 0]]] <= Divide[1,2]*Pi
Failure Failure
Failed [174 / 300]
Result: 2.331674280 <= 1.570796327
Test Values: {mu = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, p[0] = 1/2*3^(1/2)+1/2*I}


Result: 3.720287017 <= 1.570796327
Test Values: {mu = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, p[0] = -1/2+1/2*I*3^(1/2)}


Result: 1.852957222 <= 1.570796327
Test Values: {mu = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, p[0] = -1/2*3^(1/2)-1/2*I}


Result: 4.710057957 <= 1.570796327
Test Values: {mu = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, p[0] = -1.5}

... skip entries to safe data
Failed [211 / 300]
Result: False
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: False
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, 0], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
2.4.E14 I ⁢ ( z ) = ∫ t 0 b e - z ⁢ p ⁢ ( t ) ⁢ q ⁢ ( t ) ⁢ d t - ∫ t 0 a e - z ⁢ p ⁢ ( t ) ⁢ q ⁢ ( t ) ⁢ d t 𝐼 𝑧 superscript subscript subscript 𝑡 0 𝑏 superscript 𝑒 𝑧 𝑝 𝑡 𝑞 𝑡 𝑡 superscript subscript subscript 𝑡 0 𝑎 superscript 𝑒 𝑧 𝑝 𝑡 𝑞 𝑡 𝑡 {\displaystyle{\displaystyle I(z)=\int_{t_{0}}^{b}e^{-zp(t)}q(t)\mathrm{d}t-% \int_{t_{0}}^{a}e^{-zp(t)}q(t)\mathrm{d}t}}
I(z) = \int_{t_{0}}^{b}e^{-zp(t)}q(t)\diff{t}-\int_{t_{0}}^{a}e^{-zp(t)}q(t)\diff{t}		 

(int(exp(- zp(t))*q(t), t = a..b)) = int(exp(- zp(t))*q(t), t = t[0]..b)- int(exp(- zp(t))*q(t), t = t[0]..a)

(Integrate[Exp[- zp[t]]*q[t], {t, a, b}, GenerateConditions->None]) == Integrate[Exp[- zp[t]]*q[t], {t, Subscript[t, 0], b}, GenerateConditions->None]- Integrate[Exp[- zp[t]]*q[t], {t, Subscript[t, 0], a}, GenerateConditions->None]
Successful Successful - Successful [Tested: 300]
2.4.E18 p ⁢ ( α , t ) = 1 3 ⁢ w 3 + a ⁢ w 2 + b ⁢ w + c 𝑝 𝛼 𝑡 1 3 superscript 𝑤 3 𝑎 superscript 𝑤 2 𝑏 𝑤 𝑐 {\displaystyle{\displaystyle p(\alpha,t)=\tfrac{1}{3}w^{3}+aw^{2}+bw+c}}
p(\alpha,t) = \tfrac{1}{3}w^{3}+aw^{2}+bw+c		 

p(alpha , t) = (1)/(3)*(w)^(3)+ a*(w)^(2)+ b*w + c
 
p[\[Alpha], t] == Divide[1,3]*(w)^(3)+ a*(w)^(2)+ b*w + c
 Skipped - no semantic math Skipped - no semantic math - -
2.4.E20 q ⁢ ( α , t ) ⁢ d t d w = q ⁢ ( α , t ) ⁢ w 2 + 2 ⁢ a ⁢ w + b ∂ ⁡ p ⁢ ( α , t ) / ∂ ⁡ t 𝑞 𝛼 𝑡 derivative 𝑡 𝑤 𝑞 𝛼 𝑡 superscript 𝑤 2 2 𝑎 𝑤 𝑏 partial-derivative 𝑝 𝛼 𝑡 𝑡 {\displaystyle{\displaystyle q(\alpha,t)\frac{\mathrm{d}t}{\mathrm{d}w}=q(% \alpha,t)\frac{w^{2}+2aw+b}{\ifrac{\partial p(\alpha,t)}{\partial t}}}}
q(\alpha,t)\deriv{t}{w} = q(\alpha,t)\frac{w^{2}+2aw+b}{\ipderiv{p(\alpha,t)}{t}}		 

q(alpha , t)* diff(t, w) = q(alpha , t)*((w)^(2)+ 2*a*w + b)/(diff(p(alpha , t), t))

q[\[Alpha], t]* D[t, w] == q[\[Alpha], t]*Divide[(w)^(2)+ 2*a*w + b,D[p[\[Alpha], t], t]]
Error Failure - Skipped - Because timed out
2.4.E21 w = z - 1 / 3 ⁢ v - a 𝑤 superscript 𝑧 1 3 𝑣 𝑎 {\displaystyle{\displaystyle w=z^{-1/3}v-a}}
w = z^{-1/3}v-a		 

w = (z)^(- 1/3)* v - a
 
w == (z)^(- 1/3)* v - a
 Skipped - no semantic math Skipped - no semantic math - -
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