Error Functions, Dawson’s and Fresnel Integrals - 7.8 Inequalities

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
7.8.E1 𝖬 ⁑ ( x ) = ∫ x ∞ e - t 2 ⁒ d t e - x 2 Mills-ratio π‘₯ superscript subscript π‘₯ superscript 𝑒 superscript 𝑑 2 𝑑 superscript 𝑒 superscript π‘₯ 2 {\displaystyle{\displaystyle\mathsf{M}\left(x\right)=\frac{\int_{x}^{\infty}e^% {-t^{2}}\mathrm{d}t}{e^{-x^{2}}}}}
\MillsM@{x} = \frac{\int_{x}^{\infty}e^{-t^{2}}\diff{t}}{e^{-x^{2}}}		 

Error

Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}] == Divide[Integrate[Exp[- (t)^(2)], {t, x, Infinity}, GenerateConditions->None],Exp[- (x)^(2)]]
Missing Macro Error Failure -
Failed [3 / 3]
Result: Plus[-0.2849976548947546, Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 1.5, DirectedInfinity[1]}]]], {Rule[x, 1.5]}


Result: Plus[-0.545641360765047, Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 0.5, DirectedInfinity[1]}]]], {Rule[x, 0.5]}

... skip entries to safe data
7.8.E1 ∫ x ∞ e - t 2 ⁒ d t e - x 2 = e x 2 ⁒ ∫ x ∞ e - t 2 ⁒ d t superscript subscript π‘₯ superscript 𝑒 superscript 𝑑 2 𝑑 superscript 𝑒 superscript π‘₯ 2 superscript 𝑒 superscript π‘₯ 2 superscript subscript π‘₯ superscript 𝑒 superscript 𝑑 2 𝑑 {\displaystyle{\displaystyle\frac{\int_{x}^{\infty}e^{-t^{2}}\mathrm{d}t}{e^{-% x^{2}}}=e^{x^{2}}\int_{x}^{\infty}e^{-t^{2}}\mathrm{d}t}}
\frac{\int_{x}^{\infty}e^{-t^{2}}\diff{t}}{e^{-x^{2}}} = e^{x^{2}}\int_{x}^{\infty}e^{-t^{2}}\diff{t}		 

(int(exp(- (t)^(2)), t = x..infinity))/(exp(- (x)^(2))) = exp((x)^(2))*int(exp(- (t)^(2)), t = x..infinity)

Divide[Integrate[Exp[- (t)^(2)], {t, x, Infinity}, GenerateConditions->None],Exp[- (x)^(2)]] == Exp[(x)^(2)]*Integrate[Exp[- (t)^(2)], {t, x, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 3]
7.8.E2 1 x + x 2 + 2 < 𝖬 ⁑ ( x ) 1 π‘₯ superscript π‘₯ 2 2 Mills-ratio π‘₯ {\displaystyle{\displaystyle\frac{1}{x+\sqrt{x^{2}+2}}<\mathsf{M}\left(x\right% )}}
\frac{1}{x+\sqrt{x^{2}+2}} < \MillsM@{x}		 x β‰₯ 0 π‘₯ 0 {\displaystyle{\displaystyle x\geq 0}} 
Error

Divide[1,x +Sqrt[(x)^(2)+ 2]] < Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}]
Missing Macro Error Failure -
Failed [3 / 3]
Result: Less[0.28077640640441515, Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 1.5, DirectedInfinity[1]}]]], {Rule[x, 1.5]}


Result: Less[0.5, Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 0.5, DirectedInfinity[1]}]]], {Rule[x, 0.5]}

... skip entries to safe data
7.8.E2 𝖬 ⁑ ( x ) ≀ 1 x + x 2 + ( 4 / Ο€ ) Mills-ratio π‘₯ 1 π‘₯ superscript π‘₯ 2 4 πœ‹ {\displaystyle{\displaystyle\mathsf{M}\left(x\right)\leq\frac{1}{x+\sqrt{x^{2}% +(4/\pi)}}}}
\MillsM@{x} \leq \frac{1}{x+\sqrt{x^{2}+(4/\pi)}}		 x β‰₯ 0 π‘₯ 0 {\displaystyle{\displaystyle x\geq 0}} 
Error

Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}] <= Divide[1,x +Sqrt[(x)^(2)+(4/Pi)]]
Missing Macro Error Failure -
Failed [3 / 3]
Result: LessEqual[Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 1.5, DirectedInfinity[1]}]], 0.2961182351849971], {Rule[x, 1.5]}


Result: LessEqual[Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 0.5, DirectedInfinity[1]}]], 0.5766361194388748], {Rule[x, 0.5]}

... skip entries to safe data
7.8.E3 Ο€ 2 ⁒ Ο€ ⁒ x + 2 ≀ 𝖬 ⁑ ( x ) πœ‹ 2 πœ‹ π‘₯ 2 Mills-ratio π‘₯ {\displaystyle{\displaystyle\frac{\sqrt{\pi}}{2\sqrt{\pi}x+2}\leq\mathsf{M}% \left(x\right)}}
\frac{\sqrt{\pi}}{2\sqrt{\pi}x+2} \leq \MillsM@{x}		 x β‰₯ 0 π‘₯ 0 {\displaystyle{\displaystyle x\geq 0}} 
Error

Divide[Sqrt[Pi],2*Sqrt[Pi]*x + 2] <= Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}]
Missing Macro Error Failure -
Failed [3 / 3]
Result: LessEqual[0.24222581297045487, Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 1.5, DirectedInfinity[1]}]]], {Rule[x, 1.5]}


Result: LessEqual[0.46984109573138116, Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 0.5, DirectedInfinity[1]}]]], {Rule[x, 0.5]}

... skip entries to safe data
7.8.E3 𝖬 ⁑ ( x ) < 1 x + 1 Mills-ratio π‘₯ 1 π‘₯ 1 {\displaystyle{\displaystyle\mathsf{M}\left(x\right)<\frac{1}{x+1}}}
\MillsM@{x} < \frac{1}{x+1}		 x β‰₯ 0 π‘₯ 0 {\displaystyle{\displaystyle x\geq 0}} 
Error

Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}] < Divide[1,x + 1]
Missing Macro Error Failure -
Failed [3 / 3]
Result: Less[Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 1.5, DirectedInfinity[1]}]], 0.4], {Rule[x, 1.5]}


Result: Less[Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 0.5, DirectedInfinity[1]}]], 0.6666666666666666], {Rule[x, 0.5]}

... skip entries to safe data
7.8.E4 𝖬 ⁑ ( x ) < 2 3 ⁒ x + x 2 + 4 Mills-ratio π‘₯ 2 3 π‘₯ superscript π‘₯ 2 4 {\displaystyle{\displaystyle\mathsf{M}\left(x\right)<\frac{2}{3x+\sqrt{x^{2}+4% }}}}
\MillsM@{x} < \frac{2}{3x+\sqrt{x^{2}+4}}		 x > - 1 2 ⁒ 2 π‘₯ 1 2 2 {\displaystyle{\displaystyle x>-\tfrac{1}{2}\sqrt{2}}} 
Error

Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}] < Divide[2,3*x +Sqrt[(x)^(2)+ 4]]
Missing Macro Error Failure -
Failed [3 / 3]
Result: Less[Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 1.5, DirectedInfinity[1]}]], 0.2857142857142857], {Rule[x, 1.5]}


Result: Less[Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 0.5, DirectedInfinity[1]}]], 0.5615528128088303], {Rule[x, 0.5]}

... skip entries to safe data
7.8.E5 x 2 2 ⁒ x 2 + 1 ≀ x 2 ⁒ ( 2 ⁒ x 2 + 5 ) 4 ⁒ x 4 + 12 ⁒ x 2 + 3 superscript π‘₯ 2 2 superscript π‘₯ 2 1 superscript π‘₯ 2 2 superscript π‘₯ 2 5 4 superscript π‘₯ 4 12 superscript π‘₯ 2 3 {\displaystyle{\displaystyle\frac{x^{2}}{2x^{2}+1}\leq\frac{x^{2}(2x^{2}+5)}{4% x^{4}+12x^{2}+3}}}
\frac{x^{2}}{2x^{2}+1} \leq \frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3}		 x β‰₯ 0 π‘₯ 0 {\displaystyle{\displaystyle x\geq 0}} 
((x)^(2))/(2*(x)^(2)+ 1) <= ((x)^(2)*(2*(x)^(2)+ 5))/(4*(x)^(4)+ 12*(x)^(2)+ 3)

Divide[(x)^(2),2*(x)^(2)+ 1] <= Divide[(x)^(2)*(2*(x)^(2)+ 5),4*(x)^(4)+ 12*(x)^(2)+ 3]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
7.8.E5 x 2 ⁒ ( 2 ⁒ x 2 + 5 ) 4 ⁒ x 4 + 12 ⁒ x 2 + 3 ≀ x ⁒ 𝖬 ⁑ ( x ) superscript π‘₯ 2 2 superscript π‘₯ 2 5 4 superscript π‘₯ 4 12 superscript π‘₯ 2 3 π‘₯ Mills-ratio π‘₯ {\displaystyle{\displaystyle\frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3}\leq x% \mathsf{M}\left(x\right)}}
\frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3} \leq x\MillsM@{x}		 x β‰₯ 0 π‘₯ 0 {\displaystyle{\displaystyle x\geq 0}} 
Error

Divide[(x)^(2)*(2*(x)^(2)+ 5),4*(x)^(4)+ 12*(x)^(2)+ 3] <= x*Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}]
Missing Macro Error Failure Skip - symbolical successful subtest
Failed [3 / 3]
Result: LessEqual[0.4253731343283582, Times[14.23160375453779, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 1.5, DirectedInfinity[1]}]]], {Rule[x, 1.5]}


Result: LessEqual[0.22, Times[0.6420127083438707, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 0.5, DirectedInfinity[1]}]]], {Rule[x, 0.5]}

... skip entries to safe data
7.8.E5 x ⁒ 𝖬 ⁑ ( x ) < 2 ⁒ x 4 + 9 ⁒ x 2 + 4 4 ⁒ x 4 + 20 ⁒ x 2 + 15 π‘₯ Mills-ratio π‘₯ 2 superscript π‘₯ 4 9 superscript π‘₯ 2 4 4 superscript π‘₯ 4 20 superscript π‘₯ 2 15 {\displaystyle{\displaystyle x\mathsf{M}\left(x\right)<\frac{2x^{4}+9x^{2}+4}{% 4x^{4}+20x^{2}+15}}}
x\MillsM@{x} < \frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15}		 x β‰₯ 0 π‘₯ 0 {\displaystyle{\displaystyle x\geq 0}} 
Error

x*Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}] < Divide[2*(x)^(4)+ 9*(x)^(2)+ 4,4*(x)^(4)+ 20*(x)^(2)+ 15]
Missing Macro Error Failure Skip - symbolical successful subtest
Failed [3 / 3]
Result: Less[Times[14.23160375453779, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 1.5, DirectedInfinity[1]}]], 0.42834890965732086], {Rule[x, 1.5]}


Result: Less[Times[0.6420127083438707, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]]
Test Values: {t, 0.5, DirectedInfinity[1]}]], 0.31481481481481477], {Rule[x, 0.5]}

... skip entries to safe data
7.8.E5 2 ⁒ x 4 + 9 ⁒ x 2 + 4 4 ⁒ x 4 + 20 ⁒ x 2 + 15 < x 2 + 1 2 ⁒ x 2 + 3 2 superscript π‘₯ 4 9 superscript π‘₯ 2 4 4 superscript π‘₯ 4 20 superscript π‘₯ 2 15 superscript π‘₯ 2 1 2 superscript π‘₯ 2 3 {\displaystyle{\displaystyle\frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15}<\frac{x^% {2}+1}{2x^{2}+3}}}
\frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15} < \frac{x^{2}+1}{2x^{2}+3}		 x β‰₯ 0 π‘₯ 0 {\displaystyle{\displaystyle x\geq 0}} 
(2*(x)^(4)+ 9*(x)^(2)+ 4)/(4*(x)^(4)+ 20*(x)^(2)+ 15) < ((x)^(2)+ 1)/(2*(x)^(2)+ 3)

Divide[2*(x)^(4)+ 9*(x)^(2)+ 4,4*(x)^(4)+ 20*(x)^(2)+ 15] < Divide[(x)^(2)+ 1,2*(x)^(2)+ 3]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
7.8.E6 ∫ 0 x e a ⁒ t 2 ⁒ d t < 1 3 ⁒ a ⁒ x ⁒ ( 2 ⁒ e a ⁒ x 2 + a ⁒ x 2 - 2 ) superscript subscript 0 π‘₯ superscript 𝑒 π‘Ž superscript 𝑑 2 𝑑 1 3 π‘Ž π‘₯ 2 superscript 𝑒 π‘Ž superscript π‘₯ 2 π‘Ž superscript π‘₯ 2 2 {\displaystyle{\displaystyle\int_{0}^{x}e^{at^{2}}\mathrm{d}t<\frac{1}{3ax}% \left(2e^{ax^{2}}+ax^{2}-2\right)}}
\int_{0}^{x}e^{at^{2}}\diff{t} < \frac{1}{3ax}\left(2e^{ax^{2}}+ax^{2}-2\right)		 a > 0 , x > 0 formulae-sequence π‘Ž 0 π‘₯ 0 {\displaystyle{\displaystyle a>0,x>0}} 
int(exp(a*(t)^(2)), t = 0..x) < (1)/(3*a*x)*(2*exp(a*(x)^(2))+ a*(x)^(2)- 2)

Integrate[Exp[a*(t)^(2)], {t, 0, x}, GenerateConditions->None] < Divide[1,3*a*x]*(2*Exp[a*(x)^(2)]+ a*(x)^(2)- 2)
Error Failure - Successful [Tested: 9]
7.8.E7 ∫ 0 x e t 2 ⁒ d t < e x 2 - 1 x superscript subscript 0 π‘₯ superscript 𝑒 superscript 𝑑 2 𝑑 superscript 𝑒 superscript π‘₯ 2 1 π‘₯ {\displaystyle{\displaystyle\int_{0}^{x}e^{t^{2}}\mathrm{d}t<\frac{e^{x^{2}}-1% }{x}}}
\int_{0}^{x}e^{t^{2}}\diff{t} < \frac{e^{x^{2}}-1}{x}		 x > 0 π‘₯ 0 {\displaystyle{\displaystyle x>0}} 
int(exp((t)^(2)), t = 0..x) < (exp((x)^(2))- 1)/(x)

Integrate[Exp[(t)^(2)], {t, 0, x}, GenerateConditions->None] < Divide[Exp[(x)^(2)]- 1,x]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
7.8.E8 erf ⁑ x < 1 - e - 4 ⁒ x 2 / Ο€ error-function π‘₯ 1 4 superscript π‘₯ 2 {\displaystyle{\displaystyle\operatorname{erf}x<\sqrt{1-{\mathrm{e}^{-4x^{2}/% \pi}}}}}
\erf@@{x} < \sqrt{1-\expe^{-4x^{2}/\cpi}}		 x > 0 π‘₯ 0 {\displaystyle{\displaystyle x>0}} 
erf(x) < sqrt(1 - exp(- 4*(x)^(2)/Pi))

Erf[x] < Sqrt[1 - Exp[- 4*(x)^(2)/Pi]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
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