Bessel Functions - 10.14 Inequalities; Monotonicity

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DLMF Formula Constraints Maple Mathematica Symbolic
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10.14#Ex1 | J ν ⁡ ( x ) | ≤ 1 Bessel-J 𝜈 𝑥 1 {\displaystyle{\displaystyle|J_{\nu}\left(x\right)|\leq 1}}
|\BesselJ{\nu}@{x}| \leq 1		 ν ≥ 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\nu\geq 0,\Re(\nu+k+1)>0}} 
abs(BesselJ(nu, x)) <= 1

Abs[BesselJ[\[Nu], x]] <= 1
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
10.14#Ex2 | J ν ⁡ ( x ) | ≤ 2 - 1 2 Bessel-J 𝜈 𝑥 superscript 2 1 2 {\displaystyle{\displaystyle|J_{\nu}\left(x\right)|\leq 2^{-\frac{1}{2}}}}
|\BesselJ{\nu}@{x}| \leq 2^{-\frac{1}{2}}		 ν ≥ 1 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\nu\geq 1,\Re(\nu+k+1)>0}} 
abs(BesselJ(nu, x)) <= (2)^(-(1)/(2))

Abs[BesselJ[\[Nu], x]] <= (2)^(-Divide[1,2])
Failure Failure Successful [Tested: 2] Successful [Tested: 2]
10.14.E2 0 < J ν ⁡ ( ν ) 0 Bessel-J 𝜈 𝜈 {\displaystyle{\displaystyle 0<J_{\nu}\left(\nu\right)}}
0 < \BesselJ{\nu}@{\nu}		 ν > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\nu>0,\Re(\nu+k+1)>0}} 
0 < BesselJ(nu, nu)

0 < BesselJ[\[Nu], \[Nu]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
10.14.E2 J ν ⁡ ( ν ) < 2 1 3 3 2 3 ⁢ Γ ⁡ ( 2 3 ) ⁢ ν 1 3 Bessel-J 𝜈 𝜈 superscript 2 1 3 superscript 3 2 3 Euler-Gamma 2 3 superscript 𝜈 1 3 {\displaystyle{\displaystyle J_{\nu}\left(\nu\right)<\frac{2^{\frac{1}{3}}}{3^% {\frac{2}{3}}\Gamma\left(\tfrac{2}{3}\right)\nu^{\frac{1}{3}}}}}
\BesselJ{\nu}@{\nu} < \frac{2^{\frac{1}{3}}}{3^{\frac{2}{3}}\EulerGamma@{\tfrac{2}{3}}\nu^{\frac{1}{3}}}		 ν > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\nu>0,\Re(\nu+k+1)>0}} 
BesselJ(nu, nu) < ((2)^((1)/(3)))/((3)^((2)/(3))* GAMMA((2)/(3))*(nu)^((1)/(3)))

BesselJ[\[Nu], \[Nu]] < Divide[(2)^(Divide[1,3]),(3)^(Divide[2,3])* Gamma[Divide[2,3]]*\[Nu]^(Divide[1,3])]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
10.14.E3 | J n ⁡ ( z ) | ≤ e | ℑ ⁡ z | Bessel-J 𝑛 𝑧 superscript 𝑒 𝑧 {\displaystyle{\displaystyle|J_{n}\left(z\right)|\leq e^{|\Im z|}}}
|\BesselJ{n}@{z}| \leq e^{|\imagpart@@{z}|}		 ℜ ⁡ ( n + k + 1 ) > 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
abs(BesselJ(n, z)) <= exp(abs(Im(z)))

Abs[BesselJ[n, z]] <= Exp[Abs[Im[z]]]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.14.E4 | J ν ⁡ ( z ) | ≤ | 1 2 ⁢ z | ν ⁢ e | ℑ ⁡ z | Γ ⁡ ( ν + 1 ) Bessel-J 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript 𝑒 𝑧 Euler-Gamma 𝜈 1 {\displaystyle{\displaystyle|J_{\nu}\left(z\right)|\leq\frac{|\tfrac{1}{2}z|^{% \nu}e^{|\Im z|}}{\Gamma\left(\nu+1\right)}}}
|\BesselJ{\nu}@{z}| \leq \frac{|\tfrac{1}{2}z|^{\nu}e^{|\imagpart@@{z}|}}{\EulerGamma@{\nu+1}}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ν + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}} 
abs(BesselJ(nu, z)) <= ((abs((1)/(2)*z))^(nu)* exp(abs(Im(z))))/(GAMMA(nu + 1))

Abs[BesselJ[\[Nu], z]] <= Divide[(Abs[Divide[1,2]*z])^\[Nu]* Exp[Abs[Im[z]]],Gamma[\[Nu]+ 1]]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.14.E5 | J ν ⁡ ( ν ⁢ x ) | ≤ x ν ⁢ exp ⁡ ( ν ⁢ ( 1 - x 2 ) 1 2 ) ( 1 + ( 1 - x 2 ) 1 2 ) ν Bessel-J 𝜈 𝜈 𝑥 superscript 𝑥 𝜈 𝜈 superscript 1 superscript 𝑥 2 1 2 superscript 1 superscript 1 superscript 𝑥 2 1 2 𝜈 {\displaystyle{\displaystyle|J_{\nu}\left(\nu x\right)|\leq\frac{x^{\nu}\exp% \left(\nu(1-x^{2})^{\frac{1}{2}}\right)}{\left(1+(1-x^{2})^{\frac{1}{2}}\right% )^{\nu}}}}
|\BesselJ{\nu}@{\nu x}| \leq \frac{x^{\nu}\exp@{\nu(1-x^{2})^{\frac{1}{2}}}}{\left(1+(1-x^{2})^{\frac{1}{2}}\right)^{\nu}}		 ν ≥ 0 , 0 < x , x ≤ 1 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 0 formulae-sequence 0 𝑥 formulae-sequence 𝑥 1 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\nu\geq 0,0<x,x\leq 1,\Re(\nu+k+1)>0}} 
abs(BesselJ(nu, nu*x)) <= ((x)^(nu)* exp(nu*(1 - (x)^(2))^((1)/(2))))/((1 +(1 - (x)^(2))^((1)/(2)))^(nu))

Abs[BesselJ[\[Nu], \[Nu]*x]] <= Divide[(x)^\[Nu]* Exp[\[Nu]*(1 - (x)^(2))^(Divide[1,2])],(1 +(1 - (x)^(2))^(Divide[1,2]))^\[Nu]]
Failure Failure Successful [Tested: 3] Skip - No test values generated
10.14.E7 1 ≤ J ν ⁡ ( ν ⁢ x ) x ν ⁢ J ν ⁡ ( ν ) 1 Bessel-J 𝜈 𝜈 𝑥 superscript 𝑥 𝜈 Bessel-J 𝜈 𝜈 {\displaystyle{\displaystyle 1\leq\frac{J_{\nu}\left(\nu x\right)}{x^{\nu}J_{% \nu}\left(\nu\right)}}}
1 \leq \frac{\BesselJ{\nu}@{\nu x}}{x^{\nu}\BesselJ{\nu}@{\nu}}		 ν ≥ 0 , 0 < x , x ≤ 1 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 0 formulae-sequence 0 𝑥 formulae-sequence 𝑥 1 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\nu\geq 0,0<x,x\leq 1,\Re(\nu+k+1)>0}} 
1 <= (BesselJ(nu, nu*x))/((x)^(nu)* BesselJ(nu, nu))

1 <= Divide[BesselJ[\[Nu], \[Nu]*x],(x)^\[Nu]* BesselJ[\[Nu], \[Nu]]]
Failure Failure Successful [Tested: 3] Skip - No test values generated
10.14.E7 J ν ⁡ ( ν ⁢ x ) x ν ⁢ J ν ⁡ ( ν ) ≤ e ν ⁢ ( 1 - x ) Bessel-J 𝜈 𝜈 𝑥 superscript 𝑥 𝜈 Bessel-J 𝜈 𝜈 superscript 𝑒 𝜈 1 𝑥 {\displaystyle{\displaystyle\frac{J_{\nu}\left(\nu x\right)}{x^{\nu}J_{\nu}% \left(\nu\right)}\leq e^{\nu(1-x)}}}
\frac{\BesselJ{\nu}@{\nu x}}{x^{\nu}\BesselJ{\nu}@{\nu}} \leq e^{\nu(1-x)}		 ν ≥ 0 , 0 < x , x ≤ 1 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 0 formulae-sequence 0 𝑥 formulae-sequence 𝑥 1 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\nu\geq 0,0<x,x\leq 1,\Re(\nu+k+1)>0}} 
(BesselJ(nu, nu*x))/((x)^(nu)* BesselJ(nu, nu)) <= exp(nu*(1 - x))

Divide[BesselJ[\[Nu], \[Nu]*x],(x)^\[Nu]* BesselJ[\[Nu], \[Nu]]] <= Exp[\[Nu]*(1 - x)]
Failure Failure Successful [Tested: 3] Skip - No test values generated
10.14.E8 | J n ⁡ ( n ⁢ z ) | ≤ | z n ⁢ exp ⁡ ( n ⁢ ( 1 - z 2 ) 1 2 ) | | 1 + ( 1 - z 2 ) 1 2 | n Bessel-J 𝑛 𝑛 𝑧 superscript 𝑧 𝑛 𝑛 superscript 1 superscript 𝑧 2 1 2 superscript 1 superscript 1 superscript 𝑧 2 1 2 𝑛 {\displaystyle{\displaystyle|J_{n}\left(nz\right)|\leq\frac{\left|z^{n}\exp% \left(n(1-z^{2})^{\frac{1}{2}}\right)\right|}{\left|1+(1-z^{2})^{\frac{1}{2}}% \right|^{n}}}}
|\BesselJ{n}@{nz}| \leq \frac{\left|z^{n}\exp@{n(1-z^{2})^{\frac{1}{2}}}\right|}{\left|1+(1-z^{2})^{\frac{1}{2}}\right|^{n}}		 ℜ ⁡ ( n + k + 1 ) > 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
abs(BesselJ(n, n*z)) <= (abs((z)^(n)* exp(n*(1 - (z)^(2))^((1)/(2)))))/((abs(1 +(1 - (z)^(2))^((1)/(2))))^(n))

Abs[BesselJ[n, n*z]] <= Divide[Abs[(z)^(n)* Exp[n*(1 - (z)^(2))^(Divide[1,2])]],(Abs[1 +(1 - (z)^(2))^(Divide[1,2])])^(n)]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.14.E9 | J n ⁡ ( n ⁢ z ) | ≤ 1 Bessel-J 𝑛 𝑛 𝑧 1 {\displaystyle{\displaystyle|J_{n}\left(nz\right)|\leq 1}}
|\BesselJ{n}@{nz}| \leq 1		 n = 0 , ℜ ⁡ ( n + k + 1 ) > 0 formulae-sequence 𝑛 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle n=0,\Re(n+k+1)>0}} 
abs(BesselJ(n, n*z)) <= 1

Abs[BesselJ[n, n*z]] <= 1
Failure Failure Error Successful [Tested: 21]
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