Results of Bessel Functions: Difference between revisions

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! DLMF !! Formula !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
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| [https://dlmf.nist.gov/10.2.E1 10.2.E1] || [[Item:Q3005|<math>z^{2}\deriv[2]{w}{z}+z\deriv{w}{z}+(z^{2}-\nu^{2})w = 0</math>]] || <code>(z)^(2)* diff(w, [z$(2)])+ z*diff(w, z)+((z)^(2)- (nu)^(2))* w = 0</code> || <code>(z)^(2)* D[w, {z, 2}]+ z*D[w, z]+((z)^(2)- \[Nu]^(2))* w == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [217 / 300]<div class="mw-collapsible-content"><code>217/300]: [[-.8660254040e-9-2.000000001*I <- {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br><code>-.8660254040e-9-2.000000001*I <- {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [240 / 300]<div class="mw-collapsible-content"><code>{Complex[1.1102230246251565*^-16, 2.0] <- {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[1.1102230246251565*^-16, 2.0] <- {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</code><br></div></div>
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| [https://dlmf.nist.gov/10.2.E2 10.2.E2] || [[Item:Q3006|<math>\BesselJ{\nu}@{z} = (\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\EulerGamma@{\nu+k+1}}</math>]] || <code>BesselJ(nu, z) = ((1)/(2)*z)^(nu)* sum((- 1)^(k)*(((1)/(4)*(z)^(2))^(k))/(factorial(k)*GAMMA(nu + k + 1)), k = 0..infinity)</code> || <code>BesselJ[\[Nu], z] == (Divide[1,2]*z)^\[

Latest revision as of 16:53, 25 May 2021

Notation
10.1 Special Notation
Bessel and Hankel Functions
10.2 Definitions
10.3 Graphics
10.4 Connection Formulas
10.5 Wronskians and Cross-Products
10.6 Recurrence Relations and Derivatives
10.7 Limiting Forms
10.8 Power Series
10.9 Integral Representations
10.10 Continued Fractions
10.11 Analytic Continuation
10.12 Generating Function and Associated Series
10.13 Other Differential Equations
10.14 Inequalities; Monotonicity
10.15 Derivatives with Respect to Order
10.16 Relations to Other Functions
10.17 Asymptotic Expansions for Large Argument
10.18 Modulus and Phase Functions
10.19 Asymptotic Expansions for Large Order
10.20 Uniform Asymptotic Expansions for Large Order
10.21 Zeros
10.22 Integrals
10.23 Sums
10.24 Functions of Imaginary Order
Modified Bessel Functions
10.25 Definitions
10.26 Graphics
10.27 Connection Formulas
10.28 Wronskians and Cross-Products
10.29 Recurrence Relations and Derivatives
10.30 Limiting Forms
10.31 Power Series
10.32 Integral Representations
10.33 Continued Fractions
10.34 Analytic Continuation
10.35 Generating Function and Associated Series
10.36 Other Differential Equations
10.37 Inequalities; Monotonicity
10.38 Derivatives with Respect to Order
10.39 Relations to Other Functions
10.40 Asymptotic Expansions for Large Argument
10.41 Asymptotic Expansions for Large Order
10.42 Zeros
10.43 Integrals
10.44 Sums
10.45 Functions of Imaginary Order
10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
Spherical Bessel Functions
10.47 Definitions and Basic Properties
10.48 Graphs
10.49 Explicit Formulas
10.50 Wronskians and Cross-Products
10.51 Recurrence Relations and Derivatives
10.52 Limiting Forms
10.53 Power Series
10.54 Integral Representations
10.55 Continued Fractions
10.56 Generating Functions
10.57 Uniform Asymptotic Expansions for Large Order
10.58 Zeros
10.59 Integrals
10.60 Sums
Kelvin Functions
10.61 Definitions and Basic Properties
10.62 Graphs
10.63 Recurrence Relations and Derivatives
10.64 Integral Representations
10.65 Power Series
10.66 Expansions in Series of Bessel Functions
10.67 Asymptotic Expansions for Large Argument
10.68 Modulus and Phase Functions
10.69 Uniform Asymptotic Expansions for Large Order
10.70 Zeros
10.71 Integrals
Applications
10.72 Mathematical Applications
10.73 Physical Applications
Computation
10.74 Methods of Computation
10.75 Tables
10.76 Approximations
10.77 Software
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