{"history":[],"article":{"id":"40ed8641-3f3f-4970-bf91-79cec7e6c1ef","title":"Incomplete gamma function","slug":"incomplete-gamma-function","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **upper** and **lower incomplete gamma functions** are types of [special functions](https://bliptext.com/articles/special-functions) which arise as solutions to various mathematical problems such as certain [integrals](https://bliptext.com/articles/integral).\nTheir respective names stem from their integral definitions, which are defined similarly to the [gamma function](https://bliptext.com/articles/gamma-function) but with different or \"incomplete\" integral limits.\nThe gamma function is defined as an integral from zero to infinity.\nThis contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit.\nSimilarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity.\n# [The upper incomplete gamma function for some values of s: 0 (blue), 1 (red), 2 (green), 3 (orange), 4 (purple). | upload.wikimedia.org/wikipedia/commons/8/8e/Upper///incomplete///gamma///function.jpg]\n# [Plot of the regularized incomplete gamma function Q(2,z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D | upload.wikimedia.org/wikipedia/commons/d/da/Plot///of///the///regularized///incomplete///gamma///function///Q%282%2Cz%29///in///the///complex///plane///from///-2-2i///to///2%2B2i///with///colors///created///with///Mathematica///13.1///function///ComplexPlot3D.svg]\n\n# Definition\nThe upper incomplete gamma function is defined as:\nwhereas the lower incomplete gamma function is defined as:\nIn both cases s is a complex parameter, such that the real part of s is positive.\n\n# Properties\nBy [integration by parts](https://bliptext.com/articles/integration-by-parts) we find the [recurrence relations](https://bliptext.com/articles/recurrence-relations)\nand\nSince the ordinary gamma function is defined as\nwe have\nand\n\n## Continuation to complex values\nThe lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive s and x, can be developed into [holomorphic functions](https://bliptext.com/articles/holomorphic-function), with respect both to x and s, defined for almost all combinations of complex x and s. Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts.\n\n### Lower incomplete gamma function\n\n#### Holomorphic extension\nRepeated application of the recurrence relation for the **lower incomplete gamma** function leads to the [power series](https://bliptext.com/articles/power-series) expansion:\nGiven the rapid growth in [absolute value](https://bliptext.com/articles/absolute-value) of\nΓ(z + k)\nwhen\nk → ∞\n, and the fact that the [[Reciprocal Gamma function|reciprocal of\nΓ(z)\n]] is an [entire function](https://bliptext.com/articles/entire-function), the coefficients in the rightmost sum are well-defined, and locally the sum [converges uniformly](https://bliptext.com/articles/uniform-convergence) for all complex s and x. By a theorem of [Weierstrass](https://bliptext.com/articles/weierstrass), the limiting function, sometimes denoted as \\gamma^*,\nis [entire](https://bliptext.com/articles/entire-function) with respect to both z (for fixed s) and s (for fixed z), and, thus, holomorphic on\nC × C\nby [Hartog's theorem](https://bliptext.com/articles/hartog-s-theorem).\nHence, the following decomposition\nextends the real lower incomplete gamma function as a [holomorphic function](https://bliptext.com/articles/holomorphic-function), both jointly and separately in z and s. It follows from the properties of z^s and the [Γ-function](https://bliptext.com/articles/gamma-function), that the first two factors capture the [singularities](https://bliptext.com/articles/mathematical-singularity) of \\gamma(s,z) (at\nz = 0\nor s a non-positive integer), whereas the last factor contributes to its zeros.\n\n#### Multi-valuedness\nThe [complex logarithm](https://bliptext.com/articles/complex-logarithm)\n+ i arg z\nis determined up to a multiple of\n2πi\nonly, which renders it [multi-valued](https://bliptext.com/articles/multi-valued-function).\nFunctions involving the complex logarithm typically inherit this property.\nAmong these are the [complex power](https://bliptext.com/articles/exponentiation), and, since\nzs\nappears in its decomposition, the\nγ\n-function, too.\nThe indeterminacy of multi-valued functions introduces complications, since it must be stated how to select a value.\nStrategies to handle this are:\nC\nof multi-valued functions by a suitable manifold in\nC × C\ncalled [Riemann surface](https://bliptext.com/articles/riemann-surface).\nWhile this removes multi-valuedness, one has to know the theory behind it;\nThe following set of rules can be used to interpret formulas in this section correctly.\nIf not mentioned otherwise, the following is assumed:\n\n##### Sectors\nSectors in\nC\nhaving their vertex at\nz = 0\noften prove to be appropriate domains for complex expressions.\nA sector D consists of all complex z fulfilling\nz ≠ 0\nand\nα − δ < arg z < α + δ\nwith some α and\n0 < δ ≤ π\n. Often, α can be arbitrarily chosen and is not specified then.\nIf δ is not given, it is assumed to be π, and the sector is in fact the whole plane\nC\n, with the exception of a half-line originating at\nz = 0\nand pointing into the direction of\n−α\n, usually serving as a [branch cut](https://bliptext.com/articles/branch-cut).\nNote: In many applications and texts, α is silently taken to be 0, which centers the sector around the positive real axis.\n\n##### Branches\nIn particular, a single-valued and holomorphic logarithm exists on any such sector D having its imaginary part bound to the range.\nBased on such a restricted logarithm,\nzs\nand the incomplete gamma functions in turn collapse to single-valued, holomorphic functions on D (or\nC×D\n), called branches of their multi-valued counterparts on D. Adding a multiple of\n2π\nto α yields a different set of correlated branches on the same set D. However, in any given context here, α is assumed fixed and all branches involved are associated to it.\nIf\n, the branches are called [principal](https://bliptext.com/articles/principal-branch), because they equal their real analogues on the positive real axis.\nNote: In many applications and texts, formulas hold only for principal branches.\n\n##### Relation between branches\nThe values of different branches of both the complex power function and the lower incomplete gamma function can be derived from each other by multiplication of, for k a suitable integer.\n\n#### Behavior near branch point\nThe decomposition above further shows, that γ behaves near\nz = 0\n[asymptotically](https://bliptext.com/articles/asymptotic) like:\nFor positive real x, y and s,\nxy/y → 0\n, when\n(x, y) → (0, s)\n. This seems to justify setting\nγ(s, 0) = 0\nfor real\ns > 0\n. However, matters are somewhat different in the complex realm.\nOnly if (a) the real part of s is positive, and (b) values\nuv\nare taken from just a finite set of branches, they are guaranteed to converge to zero as\n(u, v) → (0, s)\n, and so does\nγ(u, v)\n. On a single [branch](https://bliptext.com/articles/branch-point) of\nγ(b)\nis naturally fulfilled, so **there**\nγ(s, 0) = 0\nfor s with positive real part is a [continuous limit](https://bliptext.com/articles/continuous-function).\nAlso note that such a continuation is by no means an [analytic one](https://bliptext.com/articles/analytic-continuation).\n\n#### Algebraic relations\nAll algebraic relations and differential equations observed by the real\nγ(s, z)\nhold for its holomorphic counterpart as well.\nThis is a consequence of the identity theorem, stating that equations between holomorphic functions valid on a real interval, hold everywhere.\nIn particular, the recurrence relation and\n∂γ(s, z)/∂z = zs−1 e−z\nare preserved on corresponding branches.\n\n#### Integral representation\nThe last relation tells us, that, for fixed s, γ is a [primitive or antiderivative](https://bliptext.com/articles/primitive-function) of the holomorphic function\nzs−1 e−z\n. Consequently, for any complex\nu, v ≠ 0\n,\nholds, as long as the [path of integration](https://bliptext.com/articles/line-integral) is entirely contained in the domain of a branch of the integrand.\nIf, additionally, the real part of s is positive, then the limit\nγ(s, u) → 0\nfor\nu → 0\napplies, finally arriving at the complex integral definition of\nγ\nAny path of integration containing 0 only at its beginning, otherwise restricted to the domain of a branch of the integrand, is valid here, for example, the straight line connecting\n0\nand z.\n\n#### Limit for \n\nz → +∞\n\n##### Real values\nGiven the integral representation of a principal branch of\nγ\n, the following equation holds for all positive real s, x:\n\n##### s complex\nThis result extends to complex s. Assume first\n1 ≤ Re(s) ≤ 2\nand\n1 < a < b\n. Then\nwhere\nhas been used in the middle.\nSince the final integral becomes arbitrarily small if only a is large enough,\nγ(s, x)\nconverges uniformly for\nx → ∞\non the strip\n1 ≤ Re(s) ≤ 2\ntowards a holomorphic function, which must be Γ(s) because of the identity theorem.\nTaking the limit in the recurrence relation\nγ(s, x) = (s − 1) γ(s − 1, x) − xs − 1 e−x\nand noting, that lim\nxn e−x = 0\nfor\nx → ∞\nand all n, shows, that\nγ(s, x)\nconverges outside the strip, too, towards a function obeying the recurrence relation of the Γ-function.\nIt follows\nfor all complex s not a non-positive integer, x real and\nγ\nprincipal.\n\n##### Sectorwise convergence\nNow let u be from the sector\nwith some fixed δ (\nα = 0\n),\nγ\nbe the principal branch on this sector, and look at\nAs shown above, the first difference can be made arbitrarily small, if\nis sufficiently large.\nThe second difference allows for following estimation:\nwhere we made use of the integral representation of\nγ\nand the formula about\nabove.\nIf we integrate along the arc with radius\nR =\naround 0 connecting u and\n, then the last integral is\nwhere\nM = δ(cos δ)−Re s eIm sδ\nis a constant independent of u or R. Again referring to the behavior of\nxn e−x\nfor large x, we see that the last expression approaches 0 as R increases towards\n∞\n. In total we now have:\nif s is not a non-negative integer,\n0 < ε < π/2\nis arbitrarily small, but fixed, and\nγ\ndenotes the principal branch on this domain.\n\n#### Overview\nis:\nz = 0\nz ≠ 0\n, with simple poles at non-positive integers s.\n\n### Upper incomplete gamma function\nAs for the **upper incomplete gamma function**, a [holomorphic](https://bliptext.com/articles/holomorphic-function) extension, with respect to z or s, is given by\nat points\n(s, z)\n, where the right hand side exists.\nSince \\gamma is multi-valued, the same holds for \\Gamma, but a restriction to principal values only yields the single-valued principal branch of \\Gamma.\nWhen s is a non-positive integer in the above equation, neither part of the difference is defined, and a [limiting process](https://bliptext.com/articles/limit-of-a-function), here developed for\ns → 0\n, fills in the missing values.\n[Complex analysis](https://bliptext.com/articles/complex-analysis) guarantees [holomorphicity](https://bliptext.com/articles/holomorphic-function), because \\Gamma(s,z) proves to be [bounded](https://bliptext.com/articles/bounded-function) in a [neighbourhood](https://bliptext.com/articles/neighbourhood-mathematics) of that limit for a fixed z.\nTo determine the limit, the power series of \\gamma^* at\nz = 0\nis useful.\nWhen replacing e^{-x} by its power series in the integral definition of \\gamma, one obtains (assume x,s positive reals for now):\nor\nwhich, as a series representation of the entire \\gamma^* function, converges for all complex x (and all complex s not a non-positive integer).\nWith its restriction to real values lifted, the series allows the expansion:\nWhen\ns → 0\n(\\gamma is the [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant) here), hence,\nis the limiting function to the upper incomplete gamma function as\ns → 0\n, also known as the [exponential integral](https://bliptext.com/articles/exponential-integral) E_1(z).\nBy way of the recurrence relation, values of for positive integers n can be derived from this result,\nso the upper incomplete gamma function proves to exist and be holomorphic, with respect both to z and s, for all s and\nz ≠ 0\n.\nis:\nz = 0\nz = 0\n(the limit when ), but this is a continuous extension, not an [analytic one](https://bliptext.com/articles/analytic-continuation) (**does not** hold for real\ns < 0\n!);\nz ≠ 0\n.\n\n## Special values\nHere, is the [exponential integral](https://bliptext.com/articles/exponential-integral), is the generalized [exponential integral](https://bliptext.com/articles/exponential-integral), is the [error function](https://bliptext.com/articles/error-function), and is the complementary [error function](https://bliptext.com/articles/error-function),.\n\n## Asymptotic behavior\ns\n, the error of\nΓ(s, x) ~ −xs / s\nis on the order of\nO(xmin{s + 1, 0})\nif\ns ≠ −1\nand\nO(ln(x))\nif\ns = −1\n),\n\n# Evaluation formulae\nThe lower gamma function can be evaluated using the power series expansion:\nwhere is the [Pochhammer symbol](https://bliptext.com/articles/falling-and-rising-factorials).\nAn alternative expansion is\nwhere\nM\nis Kummer's [confluent hypergeometric function](https://bliptext.com/articles/confluent-hypergeometric-function).\n\n## Connection with Kummer's confluent hypergeometric function\nWhen the real part of z is positive,\nwhere has an infinite radius of convergence.\nAgain with [confluent hypergeometric functions](https://bliptext.com/articles/confluent-hypergeometric-functions) and employing Kummer's identity,\nFor the actual computation of numerical values, [Gauss's continued fraction](https://bliptext.com/articles/gauss-s-continued-fraction) provides a useful expansion:\nThis continued fraction converges for all complex z, provided only that s is not a negative integer.\nThe upper gamma function has the continued fraction\nand\n\n## Multiplication theorem\nThe following [multiplication theorem](https://bliptext.com/articles/multiplication-theorem) holds true:\n\n## Software implementation\nThe incomplete gamma functions are available in various of the [computer algebra systems](https://bliptext.com/articles/computer-algebra-system).\nEven if unavailable directly, however, incomplete function values can be calculated using functions commonly included in [spreadsheets](https://bliptext.com/articles/spreadsheet) (and computer algebra packages).\nIn [Excel](https://bliptext.com/articles/microsoft-excel), for example, these can be calculated using the [gamma function](https://bliptext.com/articles/gamma-function) combined with the [gamma distribution](https://bliptext.com/articles/gamma-distribution) function.\nThese follow from the definition of the [gamma distribution's cumulative distribution function](https://bliptext.com/articles/gamma-distribution).\nIn [Python](https://bliptext.com/articles/python-programming-language), the Scipy library provides implementations of incomplete gamma functions under scipy.special, however, it does not support negative values for the first argument.\nThe function gammainc from the mpmath library supports all complex arguments.\n\n# Regularized gamma functions and Poisson random variables\nTwo related functions are the regularized gamma functions:\nP(s,x) is the [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function) for [gamma random variables](https://bliptext.com/articles/gamma-distribution) with [shape parameter](https://bliptext.com/articles/shape-parameter) s and [scale parameter](https://bliptext.com/articles/scale-parameter) 1.\nWhen s is an integer, is the cumulative distribution function for [Poisson random variables](https://bliptext.com/articles/poisson-random-variable): If X is a random variable then\nThis formula can be derived by repeated integration by parts.\nIn the context of the [stable count distribution](https://bliptext.com/articles/stable-count-distribution), the s parameter can be regarded as inverse of Lévy's stability parameter \\alpha:\nwhere is a standard stable count distribution of shape.\nP(s,x) and Q(s, x) are implemented as and in [scipy](https://bliptext.com/articles/scipy).\n\n# Derivatives\nUsing the integral representation above, the derivative of the upper incomplete gamma function with respect to x is\nThe derivative with respect to its first argument s is given by\nand the second derivative by\nwhere the function T(m,s,x) is a special case of the [Meijer G-function](https://bliptext.com/articles/meijer-g-function)\nThis particular special case has internal closure properties of its own because it can be used to express all successive derivatives.\nIn general,\nwhere P_j^n is the [permutation](https://bliptext.com/articles/permutation) defined by the [Pochhammer symbol](https://bliptext.com/articles/pochhammer-symbol):\nAll such derivatives can be generated in succession from:\nand\nThis function T(m,s,x) can be computed from its series representation valid for |z| < 1,\nwith the understanding that s is not a negative integer or zero.\nIn such a case, one must use a limit.\nResults for |z| \\ge 1 can be obtained by [analytic continuation](https://bliptext.com/articles/analytic-continuation).\nSome special cases of this function can be simplified.\nFor example,, , where is the [Exponential integral](https://bliptext.com/articles/exponential-integral).\nThese derivatives and the function T(m,s,x) provide exact solutions to a number of integrals by repeated differentiation of the integral definition of the upper incomplete gamma function.\nFor example,\nThis formula can be further inflated or generalized to a huge class of [Laplace transforms](https://bliptext.com/articles/laplace-transform) and [Mellin transforms](https://bliptext.com/articles/mellin-transform).\nWhen combined with a [computer algebra system](https://bliptext.com/articles/computer-algebra-system), the exploitation of special functions provides a powerful method for solving definite integrals, in particular those encountered by practical engineering applications (see [Symbolic integration](https://bliptext.com/articles/symbolic-integration) for more details).\n\n# Indefinite and definite integrals\nThe following indefinite integrals are readily obtained using [integration by parts](https://bliptext.com/articles/integration-by-parts) (with the [constant of integration](https://bliptext.com/articles/constant-of-integration) omitted in both cases):\nThe lower and the upper incomplete gamma function are connected via the [Fourier transform](https://bliptext.com/articles/fourier-transform):\nThis follows, for example, by suitable specialization of.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:41:04.649Z","updated_at":"2025-01-25T19:41:04.649Z","search_vector":"'function':3 'gamma':2 'incomplet':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **upper** and **lower incomplete gamma functions** are types of [special functions](https://bliptext.com/articles/special-functions) which arise as solutions to various mathematical problems such as certain [integrals](https://bliptext.com/articles/integral).\nTheir respective names stem from their integral definitions, which are defined similarly to the [gamma function](https://bliptext.com/articles/gamma-function) but with different or \"incomplete\" integral limits.\nThe gamma function is defined as an integral from zero to infinity.\nThis contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit.\nSimilarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity.\n# [The upper incomplete gamma function for some values of s: 0 (blue), 1 (red), 2 (green), 3 (orange), 4 (purple). | upload.wikimedia.org/wikipedia/commons/8/8e/Upper///incomplete///gamma///function.jpg]\n# [Plot of the regularized incomplete gamma function Q(2,z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D | upload.wikimedia.org/wikipedia/commons/d/da/Plot///of///the///regularized///incomplete///gamma///function///Q%282%2Cz%29///in///the///complex///plane///from///-2-2i///to///2%2B2i///with///colors///cre ated///with///Mathematica///13.1///function///ComplexPlot3D.svg]\n\n# Definition\nThe upper incomplete gamma function is defined as:\nwhereas the lower incomplete gamma function is defined as:\nIn both cases s is a complex parameter, such that the real part of s is positive.\n\n# Properties\nBy [integration by parts](https://bliptext.com/articles/integration-by-parts) we find the [recurrence relations](https://bliptext.com/articles/recurrence-relations)\nand\nSince the ordinary gamma function is defined as\nwe have\nand\n\n## Continuation to complex values\nThe lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive s and x, can be developed into [holomorphic functions](https://bliptext.com/articles/holomorphic-function), with respect both to x and s, defined for almost all combinations of complex x and s. Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts.\n\n### Lower incomplete gamma function\n\n#### Holomorphic extension\nRepeated application of the recurrence relation for the **lower incomplete gamma** function leads to the [power series](https://bliptext.com/articles/power-series) expansion:\nGiven the rapid growth in [absolute value](https://bliptext.com/articles/absolute-value) of\nΓ(z + k)\nwhen\nk → ∞\n, and the fact that the [[Reciprocal Gamma function|reciprocal of\nΓ(z)\n]] is an [entire function](https://bliptext.com/articles/entire-function), the coefficients in the rightmost sum are well-defined, and locally the sum [converges uniformly](https://bliptext.com/articles/uniform-convergence) for all complex s and x. By a theorem of [Weierstrass](https://bliptext.com/articles/weierstrass), the limiting function, sometimes denoted as \\gamma^*,\nis [entire](https://bliptext.com/articles/entire-function) with respect to both z (for fixed s) and s (for fixed z), and, thus, holomorphic on\nC × C\nby [Hartog's theorem](https://bliptext.com/articles/hartog-s-theorem).\nHence, the following decomposition\nextends the real lower incomplete gamma function as a [holomorphic function](https://bliptext.com/articles/holomorphic-function), both jointly and separately in z and s. It follows from the properties of z^s and the [Γ-function](https://bliptext.com/articles/gamma-function), that the first two factors capture the [singularities](https://bliptext.com/articles/mathematical-singularity) of \\gamma(s,z) (at\nz = 0\nor s a non-positive integer), whereas the last factor contributes to its zeros.\n\n#### Multi-valuedness\nThe [complex logarithm](https://bliptext.com/articles/complex-logarithm)\n+ i arg z\nis determined up to a multiple of\n2πi\nonly, which renders it [multi-valued](https://bliptext.com/articles/multi-valued-function).\nFunctions involving the complex logarithm typically inherit this property.\nAmong these are the [complex power](https://bliptext.com/articles/exponentiation), and, since\nzs\nappears in its decomposition, the\nγ\n-function, too.\nThe indeterminacy of multi-valued functions introduces complications, since it must be stated how to select a value.\nStrategies to handle this are:\nC\nof multi-valued functions by a suitable manifold in\nC × C\ncalled [Riemann surface](https://bliptext.com/articles/riemann-surface).\nWhile this removes multi-valuedness, one has to know the theory behind it;\nThe following set of rules can be used to interpret formulas in this section correctly.\nIf not mentioned otherwise, the following is assumed:\n\n##### Sectors\nSectors in\nC\nhaving their vertex at\nz = 0\noften prove to be appropriate domains for complex expressions.\nA sector D consists of all complex z fulfilling\nz ≠ 0\nand\nα − δ 0\n. However, matters are somewhat different in the complex realm.\nOnly if (a) the real part of s is positive, and (b) values\nuv\nare taken from just a finite set of branches, they are guaranteed to converge to zero as\n(u, v) → (0, s)\n, and so does\nγ(u, v)\n. On a single [branch](https://bliptext.com/articles/branch-point) of\nγ(b)\nis naturally fulfilled, so **there**\nγ(s, 0) = 0\nfor s with positive real part is a [continuous limit](https://bliptext.com/articles/continuous-function).\nAlso note that such a continuation is by no means an [analytic one](https://bliptext.com/articles/analytic-continuation).\n\n#### Algebraic relations\nAll algebraic relations and differential equations observed by the real\nγ(s, z)\nhold for its holomorphic counterpart as well.\nThis is a consequence of the identity theorem, stating that equations between holomorphic functions valid on a real interval, hold everywhere.\nIn particular, the recurrence relation and\n∂γ(s, z)/∂z = zs−1 e−z\nare preserved on corresponding branches.\n\n#### Integral representation\nThe last relation tells us, that, for fixed s, γ is a [primitive or antiderivative](https://bliptext.com/articles/primitive-function) of the holomorphic function\nzs−1 e−z\n. Consequently, for any complex\nu, v ≠ 0\n,\nholds, as long as the [path of integration](https://bliptext.com/articles/line-integral) is entirely contained in the domain of a branch of the integrand.\nIf, additionally, the real part of s is positive, then the limit\nγ(s, u) → 0\nfor\nu → 0\napplies, finally arriving at the complex integral definition of\nγ\nAny path of integration containing 0 only at its beginning, otherwise restricted to the domain of a branch of the integrand, is valid here, for example, the straight line connecting\n0\nand z.\n\n#### Limit for \n\nz → +∞\n\n##### Real values\nGiven the integral representation of a principal branch of\nγ\n, the following equation holds for all positive real s, x:\n\n##### s complex\nThis result extends to complex s. Assume first\n1 ≤ Re(s) ≤ 2\nand\n1 < a < b\n. Then\nwhere\nhas been used in the middle.\nSince the final integral becomes arbitrarily small if only a is large enough,\nγ(s, x)\nconverges uniformly for\nx → ∞\non the strip\n1 ≤ Re(s) ≤ 2\ntowards a holomorphic function, which must be Γ(s) because of the identity theorem.\nTaking the limit in the recurrence relation\nγ(s, x) = (s − 1) γ(s − 1, x) − xs − 1 e−x\nand noting, that lim\nxn e−x = 0\nfor\nx → ∞\nand all n, shows, that\nγ(s, x)\nconverges outside the strip, too, towards a function obeying the recurrence relation of the Γ-function.\nIt follows\nfor all complex s not a non-positive integer, x real and\nγ\nprincipal.\n\n##### Sectorwise convergence\nNow let u be from the sector\nwith some fixed δ (\nα = 0\n),\nγ\nbe the principal branch on this sector, and look at\nAs shown above, the first difference can be made arbitrarily small, if\nis sufficiently large.\nThe second difference allows for following estimation:\nwhere we made use of the integral representation of\nγ\nand the formula about\nabove.\nIf we integrate along the arc with radius\nR =\naround 0 connecting u and\n, then the last integral is\nwhere\nM = δ(cos δ)−Re s eIm sδ\nis a constant independent of u or R. Again referring to the behavior of\nxn e−x\nfor large x, we see that the last expression approaches 0 as R increases towards\n∞\n. In total we now have:\nif s is not a non-negative integer,\n0 < ε < π/2\nis arbitrarily small, but fixed, and\nγ\ndenotes the principal branch on this domain.\n\n#### Overview\nis:\nz = 0\nz ≠ 0\n, with simple poles at non-positive integers s.\n\n### Upper incomplete gamma function\nAs for the **upper incomplete gamma function**, a [holomorphic](https://bliptext.com/articles/holomorphic-function) extension, with respect to z or s, is given by\nat points\n(s, z)\n, where the right hand side exists.\nSince \\gamma is multi-valued, the same holds for \\Gamma, but a restriction to principal values only yields the single-valued principal branch of \\Gamma.\nWhen s is a non-positive integer in the above equation, neither part of the difference is defined, and a [limiting process](https://bliptext.com/articles/limit-of-a-function), here developed for\ns → 0\n, fills in the missing values.\n[Complex analysis](https://bliptext.com/articles/complex-analysis) guarantees [holomorphicity](https://bliptext.com/articles/holomorphic-function), because \\Gamma(s,z) proves to be [bounded](https://bliptext.com/articles/bounded-function) in a [neighbourhood](https://bliptext.com/articles/neighbourhood-mathematics) of that limit for a fixed z.\nTo determine the limit, the power series of \\gamma^* at\nz = 0\nis useful.\nWhen replacing e^{-x} by its power series in the integral definition of \\gamma, one obtains (assume x,s positive reals for now):\nor\nwhich, as a series representation of the entire \\gamma^* function, converges for all complex x (and all complex s not a non-positive integer).\nWith its restriction to real values lifted, the series allows the expansion:\nWhen\ns → 0\n(\\gamma is the [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant) here), hence,\nis the limiting function to the upper incomplete gamma function as\ns → 0\n, also known as the [exponential integral](https://bliptext.com/articles/exponential-integral) E_1(z).\nBy way of the recurrence relation, values of for positive integers n can be derived from this result,\nso the upper incomplete gamma function proves to exist and be holomorphic, with respect both to z and s, for all s and\nz ≠ 0\n.\nis:\nz = 0\nz = 0\n(the limit when ), but this is a continuous extension, not an [analytic one](https://bliptext.com/articles/analytic-continuation) (**does not** hold for real\ns < 0\n!);\nz ≠ 0\n.\n\n## Special values\nHere, is the [exponential integral](https://bliptext.com/articles/exponential-integral), is the generalized [exponential integral](https://bliptext.com/articles/exponential-integral), is the [error function](https://bliptext.com/articles/error-function), and is the complementary [error function](https://bliptext.com/articles/error-function),.\n\n## Asymptotic behavior\ns\n, the error of\nΓ(s, x) ~ −xs / s\nis on the order of\nO(xmin{s + 1, 0})\nif\ns ≠ −1\nand\nO(ln(x))\nif\ns = −1\n),\n\n# Evaluation formulae\nThe lower gamma function can be evaluated using the power series expansion:\nwhere is the [Pochhammer symbol](https://bliptext.com/articles/falling-and-rising-factorials).\nAn alternative expansion is\nwhere\nM\nis Kummer's [confluent hypergeometric function](https://bliptext.com/articles/confluent-hypergeometric-function).\n\n## Connection with Kummer's confluent hypergeometric function\nWhen the real part of z is positive,\nwhere has an infinite radius of convergence.\nAgain with [confluent hypergeometric functions](https://bliptext.com/articles/confluent-hypergeometric-functions) and employing Kummer's identity,\nFor the actual computation of numerical values, [Gauss's continued fraction](https://bliptext.com/articles/gauss-s-continued-fraction) provides a useful expansion:\nThis continued fraction converges for all complex z, provided only that s is not a negative integer.\nThe upper gamma function has the continued fraction\nand\n\n## Multiplication theorem\nThe following [multiplication theorem](https://bliptext.com/articles/multiplication-theorem) holds true:\n\n## Software implementation\nThe incomplete gamma functions are available in various of the [computer algebra systems](https://bliptext.com/articles/computer-algebra-system).\nEven if unavailable directly, however, incomplete function values can be calculated using functions commonly included in [spreadsheets](https://bliptext.com/articles/spreadsheet) (and computer algebra packages).\nIn [Excel](https://bliptext.com/articles/microsoft-excel), for example, these can be calculated using the [gamma function](https://bliptext.com/articles/gamma-function) combined with the [gamma distribution](https://bliptext.com/articles/gamma-distribution) function.\nThese follow from the definition of the [gamma distribution's cumulative distribution function](https://bliptext.com/articles/gamma-distribution).\nIn [Python](https://bliptext.com/articles/python-programming-language), the Scipy library provides implementations of incomplete gamma functions under scipy.special, however, it does not support negative values for the first argument.\nThe function gammainc from the mpmath library supports all complex arguments.\n\n# Regularized gamma functions and Poisson random variables\nTwo related functions are the regularized gamma functions:\nP(s,x) is the [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function) for [gamma random variables](https://bliptext.com/articles/gamma-distribution) with [shape parameter](https://bliptext.com/articles/shape-parameter) s and [scale parameter](https://bliptext.com/articles/scale-parameter) 1.\nWhen s is an integer, is the cumulative distribution function for [Poisson random variables](https://bliptext.com/articles/poisson-random-variable): If X is a random variable then\nThis formula can be derived by repeated integration by parts.\nIn the context of the [stable count distribution](https://bliptext.com/articles/stable-count-distribution), the s parameter can be regarded as inverse of Lévy's stability parameter \\alpha:\nwhere is a standard stable count distribution of shape.\nP(s,x) and Q(s, x) are implemented as and in [scipy](https://bliptext.com/articles/scipy).\n\n# Derivatives\nUsing the integral representation above, the derivative of the upper incomplete gamma function with respect to x is\nThe derivative with respect to its first argument s is given by\nand the second derivative by\nwhere the function T(m,s,x) is a special case of the [Meijer G-function](https://bliptext.com/articles/meijer-g-function)\nThis particular special case has internal closure properties of its own because it can be used to express all successive derivatives.\nIn general,\nwhere P_j^n is the [permutation](https://bliptext.com/articles/permutation) defined by the [Pochhammer symbol](https://bliptext.com/articles/pochhammer-symbol):\nAll such derivatives can be generated in succession from:\nand\nThis function T(m,s,x) can be computed from its series representation valid for |z| < 1,\nwith the understanding that s is not a negative integer or zero.\nIn such a case, one must use a limit.\nResults for |z| \\ge 1 can be obtained by [analytic continuation](https://bliptext.com/articles/analytic-continuation).\nSome special cases of this function can be simplified.\nFor example,, , where is the [Exponential integral](https://bliptext.com/articles/exponential-integral).\nThese derivatives and the function T(m,s,x) provide exact solutions to a number of integrals by repeated differentiation of the integral definition of the upper incomplete gamma function.\nFor example,\nThis formula can be further inflated or generalized to a huge class of [Laplace transforms](https://bliptext.com/articles/laplace-transform) and [Mellin transforms](https://bliptext.com/articles/mellin-transform).\nWhen combined with a [computer algebra system](https://bliptext.com/articles/computer-algebra-system), the exploitation of special functions provides a powerful method for solving definite integrals, in particular those encountered by practical engineering applications (see [Symbolic integration](https://bliptext.com/articles/symbolic-integration) for more details).\n\n# Indefinite and definite integrals\nThe following indefinite integrals are readily obtained using [integration by parts](https://bliptext.com/articles/integration-by-parts) (with the [constant of integration](https://bliptext.com/articles/constant-of-integration) omitted in both cases):\nThe lower and the upper incomplete gamma function are connected via the [Fourier transform](https://bliptext.com/articles/fourier-transform):\nThis follows, for example, by suitable specialization of.","revisionCount":0},"hasMore":false,"nextCursor":null}

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