
|

|

|

|

|
|

|

|

|

|

|
|

|

|

|

|

|
|

|

|

|

|
|

|

|

|

|
|

|

|

|

|

|
http://functions.wolfram.com/06.05.06.0025.01
|
|

|

|

|

|
|
|
|

|

|

|

|
|

|

|

|

|

|
Gamma[z + \[Epsilon]] == Gamma[z] Sum[(Derivative[k][Gamma][z]/(Gamma[z] k!))
\[Epsilon]^k, {k, 0, Infinity}] /; !(Element[z, Integers] && z <= 0)
|
|

|

|

|

|
|

|

|

|

|
|

|

|

|

|

|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["z", "+", "\[Epsilon]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Gamma", "[", "z", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "z", "]"]], RowBox[List[RowBox[List["Gamma", "[", "z", "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox["\[Epsilon]", "k"]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["z", "\[Element]", "Integers"]], "\[And]", RowBox[List["z", "\[LessEqual]", "0"]]]], "]"]]]]]]
|
|

|

|

|

|
|

|

|

|

|
|

|

|

|

|

|
|

|

|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> ϵ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mi> Γ </mi> <semantics> <mrow> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "k", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ϵ </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ¬ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ≤ </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> z </ci> <ci> ϵ </ci> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> k </ci> </degree> </bvar> <apply> <ci> Gamma </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> z </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> ϵ </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <not /> <apply> <and /> <apply> <in /> <ci> z </ci> <integers /> </apply> <apply> <leq /> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|

|

|

|

|

| 
| 
| 
| 
| | 
| 
| 
| 
| 
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Gamma", "[", RowBox[List["z_", "+", "\[Epsilon]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "z", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "z", "]"]], " ", SuperscriptBox["\[Epsilon]", "k"]]], RowBox[List[RowBox[List["Gamma", "[", "z", "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["z", "\[Element]", "Integers"]], "&&", RowBox[List["z", "\[LessEqual]", "0"]]]], ")"]]]]]]]]]] |
| 
| 
| 
| 
|
|

|

|

|

|
Date Added to functions.wolfram.com (modification date)
|
|

|

|

|

|

|
|

|

|

|

|
|
 |
|
|