Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 NeumannO

Source Notebook

Evaluate the Neumann polynomial

Contributed by: Yury Brychkov and Jan Mangaldan

ResourceFunction["NeumannO"][n,z]

gives the Neumann polynomial On(z) .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
The Neumann polynomial is defined as:
Neumann polynomials satisfy the generalized generating function relation .
Neumann polynomials satisfy the integral representation .
Neumann polynomials satisfy the differential equation
Neumann polynomials are rational functions and not strictly polynomials.
ResourceFunction["NeumannO"] can be evaluated to arbitrary numerical precision.
ResourceFunction["NeumannO"] automatically threads over lists.

Examples

Basic Examples (3) 

Evaluate numerically:

In[1]:=
ResourceFunction["NeumannO"][3, 2.41]
Out[1]=

Evaluate Neumann polynomials for various orders:

In[2]:=
Table[ResourceFunction["NeumannO"][n, z], {n, -4, 4}] // Expand
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Plot with respect to z:

In[3]:=
Plot[ResourceFunction["NeumannO"][Range[0, 3], z] // Evaluate, {z, 0, 5}]
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Scope (3) 

Evaluate for complex arguments:

In[4]:=
ResourceFunction["NeumannO"][3, 2.1 I]
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Evaluate to high precision:

In[5]:=
N[ResourceFunction["NeumannO"][4, 9/2], 50]
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The precision of the output tracks the precision of the input:

In[6]:=
ResourceFunction["NeumannO"][4, 4.500000000000000000000000]
Out[6]=

NeumannO threads elementwise over lists:

In[7]:=
ResourceFunction["NeumannO"][-2, {0.5, 1.0, 1.5}]
Out[7]=

Applications (3) 

Define a function:

Use NeumannO to expand a function in a Bessel function series:

In[8]:=
(* order of approximation *) ord = 19; coef = Table[1/(\[Pi] I) Integrate[ f[z] ResourceFunction["NeumannO"][k, z], {z, -1, -I, 1, I, -1}], {k, 0, ord}]; approx = coef . BesselJ[Range[0, ord], z]
Out[8]=

Compare the function with its Bessel series approximation:

In[9]:=
Plot[{f[z], approx} // Evaluate, {z, 0, 2 \[Pi]}, PlotStyle -> {AbsoluteThickness[5], AbsoluteThickness[2]}]
Out[9]=

Properties and Relations (8) 

Derivatives of Neumann polynomials are related to the polynomials themselves via :

In[10]:=
Table[\!\( \*SubscriptBox[\(\[PartialD]\), \({z, m}\)]\(\* InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["NeumannO", "ResourceFunctionLabel", FontFamily->"Source Sans Pro"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True->GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["NeumannO", "ResourceFunctionLabel", FontFamily->"Source Sans Pro"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.1\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{"\"Documentation »\"", "https://resources.wolframcloud.com/FunctionRepository/resources/70fbe704-230e-4c55-9223-d17ae371d128/"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "NeumannO"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["NeumannO"], BoxID -> "NeumannO", Selectable->False][n, z]\)\) == 2^-m \!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(m\)]\( \*SuperscriptBox[\((\(-1\))\), \(k\)]\ Binomial[m, k] \* InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["NeumannO", "ResourceFunctionLabel", FontFamily->"Source Sans Pro"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True->GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["NeumannO", "ResourceFunctionLabel", FontFamily->"Source Sans Pro"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.1\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{"\"Documentation »\"", "https://resources.wolframcloud.com/FunctionRepository/resources/70fbe704-230e-4c55-9223-d17ae371d128/"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "NeumannO"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["NeumannO"], BoxID -> "NeumannO", Selectable->False][n - m + 2\ k, z]\)\), {m, 0, 3}, {n, 0, 3}] // Simplify
Out[10]=

Neumann polynomials satisfy the differential equation :

In[11]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/c4d86f8d-94e0-49fe-86ed-eceb741328fb"]
Out[11]=

Neumann polynomials satisfy the recurrence identity :

In[12]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/0d392399-d7b7-4caf-aec1-7a6a69281797"]
Out[12]=

The Neumann polynomials have the limiting behavior given by :

In[13]:=
Table[Limit[z^(n + 1) ResourceFunction["NeumannO"][n, z], z -> 0] == 2^(-1 + n + KroneckerDelta[n]) n!, {n, 0, 10}] // Simplify
Out[13]=

Neumann polynomials can be represented as the finite sum :

In[14]:=
Table[ResourceFunction["NeumannO"][n, z] == n/4 \!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(n\)]\( \*FractionBox[\(Gamma[ \*FractionBox[\(n + k\), \(2\)]]\), \(Gamma[ \*FractionBox[\(n - k\), \(2\)] + 1]\)] \*SuperscriptBox[\(Cos[ \*FractionBox[\(\[Pi]\), \(2\)] \((n + k)\)]\), \(2\)] \*SuperscriptBox[\(( \*FractionBox[\(2\), \(z\)])\), \(k + 1\)]\)\), {n, 1, 10}] // Simplify
Out[14]=

The Neumann polynomials can be expressed in terms of HypergeometricPFQ through the formula :

In[15]:=
Table[ResourceFunction["NeumannO"][n, z] == n^((1 - (-1)^n)/2)/z^((3 - (-1)^n)/2) HypergeometricPFQ[{-\[LeftFloor]n/2\[RightFloor], \[LeftFloor]( n + 1)/2\[RightFloor], 1}, {}, -(4/z^2)], {n, 1, 10}] // Simplify
Out[15]=

Neumann polynomials can be expressed in terms of the Lommel function:

In[16]:=
Table[ResourceFunction["NeumannO"][n, z] == Binomial[n, Mod[n, 2]]/ z ResourceFunction["LommelS"][1 - Mod[n, 2], n, z], {n, 0, 10}] // Simplify
Out[16]=

Neumann polynomials can be expressed in terms of the Schläfli polynomial:

In[17]:=
Table[ResourceFunction["NeumannO"][n, z] == 1/4 (ResourceFunction["SchlaefliS"][n - 1, z] + ResourceFunction["SchlaefliS"][n + 1, z]), {n, 0, 10}] // Simplify
Out[17]=

Version History

  • 1.0.1 – 31 August 2021
  • 1.0.0 – 06 December 2019

Source Metadata

  • Citation:
    • Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G., Higher Transcendental Functions, Vol. 2. Krieger, 32–33, 1981.
    • Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 271–284, 1966.

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