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PolyGamma

PolyGamma[z]

gives the digamma function .

PolyGamma[n,z]

gives the n derivative of the digamma function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
PolyGamma[z] is the logarithmic derivative of the gamma function, given by $TemplateBox[{z}, PolyGamma]=Gamma^'(z)/TemplateBox[{z}, Gamma]$ .
PolyGamma[n,z] is given for positive integer by $TemplateBox[{n, z}, PolyGamma2]=d^nTemplateBox[{z}, PolyGamma]/dz^n$ .
For arbitrary complex n, the polygamma function is defined by fractional calculus analytic continuation.
PolyGamma[z] and PolyGamma[n,z] are meromorphic functions of z with no branch cut discontinuities.
For certain special arguments, PolyGamma automatically evaluates to exact values.
PolyGamma can be evaluated to arbitrary numerical precision.
PolyGamma automatically threads over lists.
PolyGamma can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples (7)

Evaluate the digamma function:

Wolfram Language code: PolyGamma[5]

Evaluate the pentagamma function:

Wolfram Language code: PolyGamma[3, 5]

Evaluate the second derivative of the gamma function:

Wolfram Language code: D[Gamma[z], {z, 2}]

Plot the digamma function over a subset of the reals:

Wolfram Language code: Plot[PolyGamma[x], {x, -3, 3}]

Plot over a subset of the complexes:

Wolfram Language code: ComplexPlot3D[PolyGamma[z], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]

Series expansion at the origin:

Wolfram Language code: Series[PolyGamma[z], {z, 0, 2}]

Series expansion at Infinity:

Wolfram Language code: Series[PolyGamma[z], {z, ∞, 4}]

Series expansion at a singular point:

Wolfram Language code: Series[PolyGamma[x], {x, -1, 3}]

Scope (47)

Numerical Evaluation (7)

Evaluate numerically:

Wolfram Language code: PolyGamma[100.5]

Evaluate for integer arguments of any size:

Wolfram Language code: PolyGamma[100]

Evaluate for complex arguments and orders:

Wolfram Language code: PolyGamma[2.5 + 3I]

Wolfram Language code: PolyGamma[6 + I, 2.5 + 3I]

Evaluate to any precision:

Wolfram Language code: N[PolyGamma[22 / 10], 50]

The precision of the output tracks the precision of the input:

Wolfram Language code: PolyGamma[2.20000000000000000000000]

Evaluate PolyGamma efficiently at high precision:

Wolfram Language code: PolyGamma[2.2`500]//Timing

Wolfram Language code: PolyGamma[2.2`3000];//Timing

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Wolfram Language code: PolyGamma[Interval[{1.23, 1.24}]]

Wolfram Language code: PolyGamma[1, Interval[{3.45, 3.46}]]

Wolfram Language code: PolyGamma[1, CenteredInterval[2.5, 0.01]]

Compute average-case statistical intervals using Around:

Wolfram Language code: PolyGamma[1 / 2, Around[1 / 2, 0.01]]

Compute the elementwise values of an array:

Wolfram Language code: PolyGamma[1, {{2, 1 / 2}, {1 / 2, 2}}]

Or compute the matrix PolyGamma function using MatrixFunction:

Wolfram Language code: MatrixFunction[PolyGamma[1, #]&, {{2, 1 / 2}, {1 / 2, 2}}]//FullSimplify

Specific Values (6)

Some singular points of PolyGamma:

Wolfram Language code: {PolyGamma[0], PolyGamma[-1], PolyGamma[-2]}

Values at infinity:

Wolfram Language code: {PolyGamma[-1, Infinity], PolyGamma[0, Infinity], PolyGamma[1, Infinity], PolyGamma[2, Infinity]}

Wolfram Language code: PolyGamma[ComplexInfinity]

Find a zero of :

Wolfram Language code: xzero = N@Solve[PolyGamma[x] == 0 && 1 < x < 2, x, Reals][[1, 1, 2]]//Quiet

Wolfram Language code: Plot[PolyGamma[x], {x, -2, 4}, Epilog -> Style[Point[{xzero, PolyGamma[xzero]}], PointSize[Large], Red]]

Use FunctionExpand to expand higher-order polygamma functions:

Wolfram Language code: Table[PolyGamma[n, 1], {n, 0, 4}]

Wolfram Language code: FunctionExpand[%]

Special case:

Wolfram Language code: PolyGamma[-1, z]

Evaluate at exact arguments:

Wolfram Language code: PolyGamma[-2, 7 / 2]//FunctionExpand

Wolfram Language code: PolyGamma[1, (1/4)]//FunctionExpand

Visualization (3)

Plot the digamma function:

Wolfram Language code: Plot[PolyGamma[x], {x, -4, 4}]

Plot the real part of :

Wolfram Language code: ComplexContourPlot[Re[PolyGamma[z]], {z, -2 - 2 I, 2 + 2 I}, Contours -> 20]

Plot the imaginary part of :

Wolfram Language code: ComplexContourPlot[Im[PolyGamma[z]], {z, -2 - 2 I, 2 + 2 I}, Contours -> 20]

Plot PolyGamma for half-integer values of parameter :

Wolfram Language code: Plot[{PolyGamma[1 / 2, x], PolyGamma[3 / 2, x], PolyGamma[5 / 2, x]}, {x, 0, 2}, IconizedObject[«PlotOptions»]]

Function Properties (9)

Real domain of PolyGamma:

Wolfram Language code: FunctionDomain[PolyGamma[x], x]

Wolfram Language code: FunctionDomain[PolyGamma[n, x], {n, x}]

Approximate function ranges of PolyGamma for half-integer parameters:

Wolfram Language code:

{FunctionRange[PolyGamma[1 / 2, x], x, y], FunctionRange[PolyGamma[3 / 2, x], x, y], FunctionRange[PolyGamma[5 / 2, x], x, y]}//Quiet

PolyGamma is not an analytic function:

Wolfram Language code: FunctionAnalytic[PolyGamma[x], x]

It has both singularities and discontinuities:

Wolfram Language code: FunctionSingularities[PolyGamma[x], x]

Wolfram Language code: FunctionDiscontinuities[PolyGamma[x], x]

is a meromorphic function:

Wolfram Language code: FunctionMeromorphic[PolyGamma[z], z]

For fixed non-negative integer , is a meromorphic function of :

Wolfram Language code: Table[FunctionMeromorphic[PolyGamma[n, z], z], {n, 0, 5}]

It is not meromorphic for other values of :

Wolfram Language code: {FunctionMeromorphic[PolyGamma[-1, z], z], FunctionMeromorphic[PolyGamma[π, z], z]}

is neither nondecreasing nor nonincreasing:

Wolfram Language code: FunctionMonotonicity[PolyGamma[x], x]

is not injective:

Wolfram Language code: FunctionInjective[PolyGamma[x], x]

Wolfram Language code: Plot[{PolyGamma[x], -1}, {x, -3, 7}]

is surjective:

Wolfram Language code: FunctionSurjective[PolyGamma[x], x]

Wolfram Language code: Plot[{PolyGamma[x], -10}, {x, -3, 25}]

is neither non-negative nor non-positive:

Wolfram Language code: FunctionSign[PolyGamma[x], x]

is neither convex nor concave:

Wolfram Language code: FunctionConvexity[PolyGamma[x], x]

Differentiation (3)

First derivative of PolyGamma:

Wolfram Language code: D[PolyGamma[x], x]

Wolfram Language code: D[PolyGamma[n, x], x]

Higher derivatives of the digamma function:

Wolfram Language code: Table[D[PolyGamma[x], {x, n}], {n, 1, 3}]

Wolfram Language code: Plot[Evaluate[%], {x, -4, 4}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative"}]

Formula for the derivative:

Wolfram Language code: D[PolyGamma[x], {x, n}]

Integration (3)

Indefinite integral of PolyGamma:

Wolfram Language code: Integrate[PolyGamma[x], x]

Wolfram Language code: Integrate[PolyGamma[n, x], x]

Indefinite integral involving a power function:

Wolfram Language code: Integrate[x ^ 2 PolyGamma[x], x]

Definite integral $int_1^3TemplateBox[{x}, PolyGamma]dx$ :

Wolfram Language code: Integrate[PolyGamma[x], {x, 1, 3}]

Series Expansions (7)

Taylor expansion for the digamma function around :

Wolfram Language code: Series[PolyGamma[x], {x, 1, 4}]

Plot the first three approximations for the Euler gamma function around :

Wolfram Language code:

terms = Normal@Table[Series[PolyGamma[x], {x, 1, m}], {m, 1, 3}];
Plot[{PolyGamma[x], terms}, {x, 0, 2}]

General term in the series expansion of the digamma function:

Wolfram Language code: SeriesCoefficient[PolyGamma[x], {x, 0, n}]

Series expansion at infinities:

Wolfram Language code: Series[PolyGamma[z], {z, Infinity, 2}]

Wolfram Language code: Series[PolyGamma[z], {z, -Infinity, 2}]

Series expansion at poles:

Wolfram Language code: Series[PolyGamma[z], {z, -5, 3}]

Series expansion at a generic point:

Wolfram Language code: Series[PolyGamma[n, z], {z, z0, 2}]

Series expansion near a singularity:

Wolfram Language code: Series[PolyGamma[3, -k + ϵ], {ϵ, 0, 2}, Assumptions -> k∈Integers && k ≥ 0]

PolyGamma can be applied to a power series:

Wolfram Language code: PolyGamma[Exp[x] + O[x]^5]//FullSimplify

Function Identities and Simplifications (5)

Use FullSimplify to simplify polygamma functions:

Wolfram Language code: FullSimplify[PolyGamma[1 - z] - PolyGamma[z]]

PolyGamma identity :

Wolfram Language code: PolyGamma[0, z] == PolyGamma[z]

PolyGamma of a double argument:

Wolfram Language code: PolyGamma[2 z] == (1/2)(PolyGamma[z] + PolyGamma[z + (1/2)]) + Log[2]//FullSimplify

Other argument simplifications:

Wolfram Language code: PolyGamma[-z] == PolyGamma[z] + (1/z) + π Cot[π z]//FullSimplify

Wolfram Language code: PolyGamma[z - 1] == PolyGamma[z] - (1/z - 1)//FullSimplify

Recurrence relation:

Wolfram Language code: PolyGamma[z] == PolyGamma[z + 1] - (1/z)//FullSimplify

Wolfram Language code: FullSimplify[PolyGamma[n, z] == PolyGamma[n, z + 1] - (-1)^n n! z^-n - 1, n∈Integers && n ≥ 0]

Function Representations (4)

Digamma function definition:

Wolfram Language code: PolyGamma[z] == (D[Gamma[z], z]/Gamma[z])

Integral representation:

Wolfram Language code: PolyGamma[z] == Integrate[(Exp[-t]/t) - (1/(t + 1)^z t), {t, 0, Infinity}, Assumptions -> Re[z] > 0]

PolyGamma can be represented as a DifferenceRoot:

Wolfram Language code: DifferenceRootReduce[PolyGamma[k], k]

Wolfram Language code: DifferenceRootReduce[PolyGamma[3, k], k]

TraditionalForm formatting:

Wolfram Language code: PolyGamma[z]//TraditionalForm

Wolfram Language code: PolyGamma[n, x]//TraditionalForm

Applications (4)

Plot of the absolute value of PolyGamma over the complex plane:

Wolfram Language code: Plot3D[Abs[PolyGamma[x + I y]], {x, -5, 2}, {y, -1, 1}]

The electric field energy of a charge at a fraction of the distance between parallel conducting plates:

Wolfram Language code: Sum[(1/k) - (1/2k - 2 x) - (1/2(k - 1) + 2 x), {k, Infinity}]

Expand near the left wall:

Wolfram Language code: Series[%, {x, 0, 3}]//FullSimplify

Final speed of a rocket with discrete propulsion events:

Wolfram Language code: Sum[(Subscript[m, i] - Subscript[m, f]/n) u (1/Subscript[m, f] + (k/n)(Subscript[m, i] - Subscript[m, f])), {k, n}]

Final velocity in the limit of constant continuous propulsion:

Wolfram Language code: Assuming[Subscript[m, i] > Subscript[m, f] > 0, Series[%, {n, Infinity, 1}, Assumptions -> n > 1]//FullSimplify]

Effective confining potential in random matrix theory for a Gaussian density of states:

Wolfram Language code: v[x_] := x ^ 2 HypergeometricPFQ[{1, 1}, {2, 3 / 2}, -x ^ 2]

Wolfram Language code: Plot[v[x], {x, 0, 5}]

Expansion at infinity reveals logarithmic growth:

Wolfram Language code: Series[v[x], {x, ∞, 3}]//Expand

Properties & Relations (7)

Use FullSimplify to simplify polygamma functions:

Wolfram Language code: FullSimplify[PolyGamma[1 - z] - PolyGamma[z]]

Express rational arguments through elementary functions:

Wolfram Language code: FunctionExpand[PolyGamma[1 / 5]]

Numerically find a root of a transcendental equation:

Wolfram Language code: FindRoot[PolyGamma[z] - 2 PolyGamma[z + 2] + z == 2, {z, 1}]

Sums and integrals:

Wolfram Language code: Underoverscript[∑, k = 1, n]PolyGamma[k]

Wolfram Language code: Underoverscript[∑, k = 1, n]PolyGamma[k]z ^ k

Wolfram Language code: Integrate[x ^ 2 PolyGamma[x], x]

Generate PolyGamma from integrals, sums, and limits:

Wolfram Language code: Integrate[(1 - t^z - 1/1 - t), {t, 0, 1}, Assumptions -> Re[z] > 0]

Wolfram Language code: Integrate[(Exp[-t]/t) - (1/(t + 1)^z t), {t, 0, Infinity}, Assumptions -> Re[z] > 0]

Wolfram Language code: Sum[(k^2 - 1/k^2 + 1), {k, 1, n}]

Wolfram Language code: Product[(1 + (y/x + n)) Exp[-(y/x + n)], {n, Infinity}]

Wolfram Language code: Limit[HurwitzZeta[s, z] - (1/s - 1), s -> 1]

Generating function:

Wolfram Language code: Underoverscript[∑, k = 1, ∞]PolyGamma[k] x^k

Obtain as special cases of hypergeometric functions:

Wolfram Language code: HypergeometricPFQ[{1, 1, 1, 2 - z}, {2, 2, 2}, 1]

Possible Issues (3)

The one‐argument form evaluates to the two-argument form:

Wolfram Language code: PolyGamma[z]

Large orders can give results too large to be computed explicitly:

Wolfram Language code: PolyGamma[10. ^ 16, 1]

Machine‐number inputs can give high‐precision results:

Wolfram Language code: PolyGamma[10. ^ 3, 1]

Wolfram Language code: MachineNumberQ[%]

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PolyGamma

Details

Examples

Basic Examples (7)

Scope (47)

Numerical Evaluation (7)

Specific Values (6)

Visualization (3)

Function Properties (9)

Differentiation (3)

Integration (3)

Series Expansions (7)

Function Identities and Simplifications (5)

Function Representations (4)

Applications (4)

Properties & Relations (7)

Possible Issues (3)

Text

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APA

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PolyGamma

Details

Examples

Basic Examples (7)

Scope (47)

Numerical Evaluation (7)

Specific Values (6)

Visualization (3)

Function Properties (9)

Differentiation (3)

Integration (3)

Series Expansions (7)

Function Identities and Simplifications (5)

Function Representations (4)

Applications (4)

Properties & Relations (7)

Possible Issues (3)

See Also

Tech Notes

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History

Text

CMS

APA

BibTeX

BibLaTeX