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ooooo`2Moooo0P000:[oool001coool00`000?ooooooo`2Noooo00<0003oooooool0Z?ooo`007?oo o`030000oooooooo0?oooom:oooo000Loooo0P000?oooom;oooo000Loooo00<0003oooooool0oooo od[oool001coool00`000?ooooooo`3oooooB_ooo`007?ooo`030000oooooooo0?oooom:oooo003o ooooJOooo`00ooooofWoool00?oooomYoooo003oooooJOooo`00ooooofWoool00?oooomYoooo0000 \ \>"], ImageRangeCache->{{{95.8125, 382.813}, {274.75, 97.8125}} -> {-1.94425, \ 0.188164, 0.0157891, 0.0200183}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output", CellLabel->"Out[5]="] }, Open ]], Cell["\<\ The right way of applying the Newton-Leibniz theorem is to take \ into account an influence of the jump\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \ \)\(Limit[int, \ x -> 4, \ Direction\ -> \ \ \ \ 1]\ - Limit[int, \ x -> 2, \ Direction\ -> \ \(-1\)]\ + \ Limit[int, \ x -> 2, \ Direction\ -> \ \ \ \ 1]\ - \ Limit[int, \ x -> 0, \ Direction\ -> \ \(-1\)]\)\)\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \(TraditionalForm \`\[Pi] - \(tan\^\(-1\)\)(1\/4) - \(tan\^\(-1\)\)(5\/12)\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " evaluates definite integrals in precisely that way." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_0\%4 \(\( x\^2 + 2\ x + 4\)\/\(x\^4 - 7\ x\^2 + 2\ x + 17\)\) \[DifferentialD]x\)], "Input", CellLabel->"In[7]:=", TextAlignment->Left, TextJustification->0], Cell[BoxData[ \(TraditionalForm \`\[Pi] - \(tan\^\(-1\)\)(1\/4) - \(tan\^\(-1\)\)(5\/12)\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell["\<\ The origin of discontinuities of antiderivatives along the path of \ integration is not in the method of indefinite integration but rather in the \ integrand. In the discussed example, the integrand has four singular poles \ that become branch points for the antiderivative. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NRoots[x\^4 - 7\ x\^2 + 2\ x + 17 == 0, x]\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \(TraditionalForm \`x == \(-1.95333607887442219`\) - 0.244027663540563244`\ \[ImaginaryI] \[Or] x == \(-1.95333607887442219`\) + 0.244027663540563244`\ \[ImaginaryI] \[Or] x == \(1.95333607887442219`\[InvisibleSpace]\) - 0.755972336459437155`\ \[ImaginaryI] \[Or] x == \(1.95333607887442219`\[InvisibleSpace]\) + 0.755972336459437155`\ \[ImaginaryI]\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[TextData[{ "Connected in pairs these points make two branch cuts. And the path of \ integration crosses one of them. 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oooooooo00Ooool00`000?ooooooo`06oooo00<0003oooooool01oooo`030000oooooooo00Koool0 0`000?ooooooo`07oooo00<0003oooooool01_ooo`030000oooooooo00Koool00`000?ooooooo`07 oooo00<0003oooooool01_ooo`030000oooooooo00Ooool00`000?ooooooo`06oooo00<0003ooooo ool01oooo`030000oooooooo00Koool00`000?ooooooo`07oooo00<0003oooooool01_ooo`030000 oooooooo00Koool00`000?ooooooo`07oooo00<0003oooooool01_ooo`030000oooooooo00Ooool0 0`000?ooooooo`06oooo00<0003oooooool01oooo`030000oooooooo00Koool00`000?ooooooo`06 oooo00<0003oooooool01oooo`030000oooooooo00Koool00`000?ooooooo`07oooo00<0003ooooo ool01_ooo`030000oooooooo00Ooool00`000?ooooooo`06oooo00<0003oooooool01oooo`030000 oooooooo00Koool00`000?ooooooo`06oooo00<0003oooooool01oooo`030000oooooooo00Koool0 0`000?ooooooo`07oooo00<0003oooooool00oooo`400000013ooooo0000F@000000ooooofWoool0 0001\ \>"], ImageRangeCache->{{{95.8125, 382.813}, {411.438, 124.438}} -> {-6.12213, \ -0.177356, 0.026816, 0.026816}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] ContourGraphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output", CellLabel->"Out[9]="] }, Open ]], Cell[TextData[{ "We see that in a complex plane of the variable ", StyleBox["x ", FontSlant->"Italic"], "the antiderivative has two branch cuts (bold black vertical lines) and the \ path of integration, the line (0,4), intersects the right branch cut. \ Obviously, by varying the constant of integration we can change the form of \ the antiderivative so that we would get various forms of branch cuts. Here we \ understand the constant of integration as a function f(x) such that ", Cell[BoxData[ FormBox[ FractionBox[\(df(x)\), "dx", MultilineFunction->None], TraditionalForm]]], " is zero. As a simple example let us consider the step-wise constant \ function ", Cell[BoxData[ \(\@x\^2\/x\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\[PartialD]\_x\(\@ x\^2\/x\)]\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \(TraditionalForm\`0\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell["\<\ Thinking hard, we can built an antiderivative that does not have a \ branch cut crossing a given interval of integration. However, we can never \ get rid of branch cuts!\ \>", "Text"], Cell[TextData[{ "Analysis of the singularities of antiderivatives is a time consuming and \ sometimes heuristic process, especially if trigonometric or special functions \ are involved in antiderivatives. In the latter case ", StyleBox["Integrate", FontFamily->"Courier", FontWeight->"Bold"], " may not be able to detect all singular points on the interval of \ integration, which will result in a warning message" }], "Text"], Cell[BoxData[ \(TraditionalForm\`Integrate::"gener" : \ "Unable to check convergence"\)], "Message"], Cell["\<\ You should pay attention to the message since it warns you that the \ result of the integration might be wrong.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Improper Integrals", "Section"], Cell[TextData[{ "It is quite clear that the above procedure cannot cover the whole variety \ of definite integrals. There are two reasons behind that. First, the \ correspondent indefinite integral cannot be expressed in finite terms of \ functions represented in ", StyleBox["Mathematica.", FontSlant->"Italic"], " For instance," }], "Text", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]Cos[Sin[x]] \[DifferentialD]x\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(TraditionalForm\`\[Integral]\(cos(sin(x))\) \[DifferentialD]x\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell["\<\ However, the definite integral with the specific limits of \ integration is doable.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_0\%\[Pi] Cos[Sin[x]] \[DifferentialD]x\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(TraditionalForm\`\[Pi]\ \(\(J\_0\)(1)\)\)], "Output", CellLabel->"Out[12]="] }, Open ]], Cell["\<\ Second, even if an indefinite integral can be done, it requires a \ great deal of effort to find limits at the end points. Here is an \ example,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_0\%\(\[Pi]\/2\)\(Tan[x]\^\(1/\[Pi]\)\) \[DifferentialD]x\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ \(TraditionalForm\`1\/2\ \[Pi]\ \(sec(1\/2)\)\)], "Output", CellLabel->"Out[13]="] }, Open ]], Cell["\<\ This is an improper integral since the top limit is a singular \ point of the integrand. The result of indefinite integration is\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\(Tan[x]\^\(1/\[Pi]\)\) \[DifferentialD]x\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"\[Pi]", " ", TagBox[ RowBox[{\(\(\[ThinSpace]\_2\)F\_1\), "(", RowBox[{ RowBox[{ TagBox[\(\(1 + \[Pi]\)\/\(2\ \[Pi]\)\), (Editable -> True)], ",", TagBox["1", (Editable -> True)]}], ";", TagBox[\(1 + \(1 + \[Pi]\)\/\(2\ \[Pi]\)\), (Editable -> True)], ";", TagBox[\(-\(\(tan\^2\)(x)\)\), (Editable -> True)]}], ")"}], InterpretTemplate[ Hypergeometric2F1[ #, #2, #3, #4]&]], " ", \(\(tan\^\(1 + 1\/\[Pi]\)\)(x)\)}], \(1 + \[Pi]\)], TraditionalForm]], "Output", CellLabel->"Out[14]="] }, Open ]], Cell[TextData[{ "To find the limit at ", Cell[BoxData[ \(TraditionalForm\`x = \[Pi]\/2\)]], " one has to, first, develop the asymptotic expansion of the hypergeometric \ Gauss function at \[Infinity], and second, construct the asymptotic scale for \ the Puiseux series. Currently, ", StyleBox["Mathematica'", FontSlant->"Italic"], "s ", StyleBox["Series", FontFamily->"Courier", FontWeight->"Bold"], " structure is based on power series that allow only rational exponents." }], "Text"], Cell["\<\ Following we present a short description of the algorithm for \ evaluating improper integrals. The overall idea has been given in [1] and \ [3]. Some practical details regarding \"logarithmic\" cases are described in \ [2] and [4].\ \>", "Text"], Cell[CellGroupData[{ Cell["The General Idea", "Subsection"], Cell[TextData[{ "By means of the Mellin integral transform and Parseval's equality, a given \ improper integral is transformed to a contour integral over the straight line \ ", Cell[BoxData[ \(TraditionalForm\`\((\[Gamma] - \[ImaginaryI]\[Infinity], \ \[Gamma] + \ \[ImaginaryI]\[Infinity])\)\)]], " in a complex plane of the parameter", StyleBox[" z", FontSlant->"Italic"], ":" }], "Text"], Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%\[Infinity]\(\( f\_1\)(x)\)\ \(\(f\_2\)(z\/x)\) \[DifferentialD]x\/x = \(1\/\(2\ \[Pi]\ \[ImaginaryI]\)\) \(\[Integral]\+\(\[Gamma] - \[ImaginaryI]\[Infinity]\)\%\(\[Gamma] + \[ImaginaryI]\[Infinity]\)\(\(f\_1\^*\)(s)\)\ \(\(f\_2\^*\)(s)\)\ \(z\^\(-s\)\) \[DifferentialD]s\)\)], "NumberedEquation", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\(\(f\_1\%*\)(x)\)\ and\ \(\(f\_2\%*\)(x)\)\)]], " are Mellin transforms of ", Cell[BoxData[ \(TraditionalForm\`\(\(f\_1\)(x)\)\ \ and\ \(\(f\_2\)(x)\)\)]] }], "Text"], Cell[BoxData[ \(TraditionalForm \`\(f\^*\)(s) = \[Integral]\_0\%\[Infinity]\(\( f(x)\)\ x\^\(s - 1\)\) \[DifferentialD]x\)], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "The real parameter \[Gamma] in the formula (1) is defined by conditions of \ existence of Mellin transforms ", Cell[BoxData[ \(TraditionalForm\`\(\(f\_1\%*\)(x)\)\ and\ \(\(f\_2\%*\)(x)\)\)]], ". Finally, the residue theorem is used to evaluate the contour integral in \ the right side of the formula (1)" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\(1\/\(2\ \[Pi]\ \[ImaginaryI]\)\) \(\[ContourIntegral]\+\ \[CapitalGamma]\( f(z)\) \[DifferentialD]z\)\), " ", "=", " ", FormBox[\(\[Sum]\+\(k = 1\)\%m res\+\(z\ = \ a\_k\)\ \(f(z)\)\), "TraditionalForm"]}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "where \[CapitalGamma] is a closed contour, and ", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], " are poles of ", StyleBox["f(z) ", FontSlant->"Italic"], "that", " lie in a domain bounded by", StyleBox[" \[CapitalGamma]. ", FontSlant->"Italic"], "The success of this scheme depends on two factors: first, the Mellin image \ of an integrand ", Cell[BoxData[ \(TraditionalForm\`\(\(f\_1\)(x)\)\ \(\(f\_2\)(z\/x)\)\)]], " must exist, and, second, it should be represented in terms of Gamma \ functions. If these conditions are satisfied then the contour integral in \ (1), called the Mellin-Barnes integral or the Meijer G-function, can almost \ always be expressed in finite terms of hypergeometric functions. This fact is \ known as Slater's theorem (see [1] and [5]). We said \"almost always\", \ since there is a special case of the G-function when the latter cannot be \ reduced to hypergeometric functions but to their derivatives with respect to \ parameters. By analogy with linear differential equations with polynomial \ coefficients, such a singular case of the G-function is named a logarithmic \ case. The modified Bessel function ", Cell[BoxData[ \(TraditionalForm\`\(K\_0\)(z)\)]], " is such a classical example, since its series representation" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\(K\_0\)(z)\), "=", RowBox[{\(\(-\(\(I\_0\)(z)\)\)\ \(log(z\/2)\)\), "+", RowBox[{\(\[Sum]\+\(k = 0\)\%\[Infinity]\), RowBox[{ FractionBox[ RowBox[{ TagBox["\[Psi]", PolyGamma], \((k + 1)\)}], \(\(k!\)\^2\)], " ", \(\((z\^2\/4)\)\^k\)}]}]}]}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0], Cell["cannot be expressed via hypergeometrics.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Mellin-Barnes Integrals", "Subsection"], Cell["These integrals are defined by (see[6])", "Text"], Cell[BoxData[ \(TraditionalForm \`\(1\/\(2\ \[Pi]\ \[ImaginaryI]\)\) \(\[ContourIntegral]\+\[ScriptCapitalL]\ \(g(s)\) \(z\^\(-s\)\) \[DifferentialD]s\)\)], "Text", TextAlignment->Center, TextJustification->0], Cell["where", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(g\ \((s)\)\), "=", StyleBox[ \(\(\[Product]\+\(j = 1\)\%n\ \[CapitalGamma]\ \((a\_j + s)\)\ \(\[Product]\+\(j = 1\)\%m\ \[CapitalGamma]\ \((b\_j - s)\)\)\)\/\(\[Product]\+\(j = 1 \)\%k\(\[CapitalGamma](c\_j + s)\)\ \(\[Product]\+\(j = 1\)\%l\ \[CapitalGamma](d\_j - s)\)\)\), UnderoverscriptBoxOptions->{LimitsPositioning->True}]}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "and the contour \[ScriptCapitalL] is a line that separates poles of ", Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma]\ \((a\_j + s)\)\)]], " from ", Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma]\ \((b\_j - s)\)\)]], ". Let us investigate when the integral exists. On a straight line ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL], \( = \((\[Gamma] - \[ImaginaryI]\[Infinity], \ \[Gamma] + \[ImaginaryI]\[Infinity])\)\)\)]], " the real part of ", Cell[BoxData[ \(TraditionalForm\`s \[Element] \[ScriptCapitalL]\)]], " is bounded, and ", Cell[BoxData[ \(TraditionalForm \`\[LeftBracketingBar]Im(s)\[RightBracketingBar] \[Rule] \[Infinity]\)]], ". Using the Stirling asymptotic formula " }], "Text"], Cell[BoxData[ \(TraditionalForm \`\[LeftBracketingBar]\[CapitalGamma](x + \[ImaginaryI]\ y) \[RightBracketingBar] = \@\(2\ \[Pi]\)\ \[LeftBracketingBar]y\[RightBracketingBar]\^\(x - 1\/2\)\ \(\(\[ExponentialE]\^\(\(-\(\[Pi]\/2\)\)\ \[LeftBracketingBar]y\[RightBracketingBar]\ \)\)( 1\ + \ O(1\/\[LeftBracketingBar]y\[RightBracketingBar]))\), \ \[LeftBracketingBar]y\[RightBracketingBar]\ \[Rule] \[Infinity]\)], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "we can deduce that the integrand ", Cell[BoxData[ \(TraditionalForm\`\(g(s)\) z\^\(-s\)\)]], " vanishes exponentially as \[LeftBracketingBar]Im(", StyleBox["s", FontSlant->"Italic"], ")\[RightBracketingBar]\[Rule]\[Infinity] if ", Cell[BoxData[ \(TraditionalForm\`m + n - k - l > 0\)]], ", and ", Cell[BoxData[ FormBox[ RowBox[{\(\[LeftBracketingBar]arg(z)\[RightBracketingBar]\), "<", RowBox[{\(\[Pi]\/2\), " ", RowBox[{"(", RowBox[{ StyleBox["m", FontSlant->"Italic"], "+", StyleBox["n", FontSlant->"Italic"], "-", StyleBox["k", FontSlant->"Italic"], "-", StyleBox["l", FontSlant->"Italic"]}], ")"}]}]}], TraditionalForm]]], ". If ", Cell[BoxData[ \(TraditionalForm\`m + n - k - l = 0\)]], ", then ", StyleBox["z", FontSlant->"Italic"], " must be real and positive. Some additional conditions are required here \ (see details in [1]). " }], "Text"], Cell[TextData[{ "It is clear that we can not simply imply the residue theorem to this \ contour integral. We need initially to transform the contour \ \[ScriptCapitalL] to the closed one. There are two possibilities: we can \ either transform \[ScriptCapitalL] into the left loop ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\_\(\(\ \)\(-\[Infinity]\)\)\)]], ", the contour encircling all poles of ", Cell[BoxData[ \(TraditionalForm\`\(\(\[CapitalGamma]\ \((a\_j + s)\)\)\(,\)\)\)]], " or to the right loop ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\_\(\(\ \)\(+\[Infinity]\)\)\)]], ", the contour encircling the poles of ", Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma]\ \((b\_j - s)\)\)]], ". The criteria of which contour should be chosen appears from the \ convergence of the integral along that contour. On the left-hand loop ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\_\(\(\ \)\(-\[Infinity]\)\)\)]], ", the imaginary part of ", Cell[BoxData[ FormBox[ RowBox[{"s", "\[Element]", FormBox[\(\[ScriptCapitalL]\_\(\(\ \)\(-\[Infinity]\)\(\ \)\)\), "TraditionalForm"]}], TraditionalForm]]], "is bounded and ", Cell[BoxData[ \(TraditionalForm\`Re(s) \[Rule] \(-\[Infinity]\)\)]], ". Assuming that ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]arg(\(-s\))\[RightBracketingBar] \ < \[Pi]\/2\)]], ", and making use of the Stirling formula" }], "Text"], Cell[BoxData[ \(TraditionalForm \`\[CapitalGamma](z) = \@\(2\ \[Pi]\)\ z\^\(z - 1\/2\)\ \[ExponentialE]\^\(-z\)\ \((1\ + \ O(1\/z))\), \ z \[Rule] \[Infinity], \ \[LeftBracketingBar]arg(z)\[RightBracketingBar] < \[Pi]\)], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "and the reflection formula of the Gamma function, we find that the \ Mellin-Barnes integral over the loop ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\_\(\ \(-\[Infinity]\)\)\)]], " exists, if ", Cell[BoxData[ \(TraditionalForm\`n + l - m - k > 0\)]], ". If ", Cell[BoxData[ \(TraditionalForm\`n + l - m - k = 0\)]], ", then z must be within the unit disk ", Cell[BoxData[ \(TraditionalForm\`\(|z | < 1\)\)]], ". If z is on a unit circle ", Cell[BoxData[ \(TraditionalForm\`\(|z | \)\ = \ 1\)]], ", the integral converges if " }], "Text"], Cell[BoxData[ \(TraditionalForm \`Re(\[Sum]\+\(j = 1\)\%n a\_j + \[Sum]\+\(j = 1\)\%m b\_j + \[Sum]\+\(j = 1\)\%k c\_j + \[Sum]\+\(j = 1\)\%l d\_j) < \(-k\) + n - 1\)], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "On the right-hand loop ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\_\(\ \(+\[Infinity]\)\)\)]], ", the imaginary part of ", Cell[BoxData[ FormBox[ RowBox[{"s", "\[Element]", FormBox[\(\[ScriptCapitalL]\_\(\ \(+\[Infinity]\)\ \)\), "TraditionalForm"]}], TraditionalForm]]], "is bounded and ", Cell[BoxData[ \(TraditionalForm\`Re(s) \[Rule] \(+\[Infinity]\)\)]], ". Proceeding similarly to the above, we find that the Mellin-Barnes \ integral over the loop ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\_\(\ \(+\[Infinity]\)\)\)]], " exists, if ", Cell[BoxData[ \(TraditionalForm\`n + l - m - k < 0\)]], ". If ", Cell[BoxData[ \(TraditionalForm\`n + l - m - k = 0\)]], ", then ", StyleBox["z", FontSlant->"Italic"], " must be outside of the unit disk. Additional conditions are required if ", StyleBox["z", FontSlant->"Italic"], " is on a unit circle." }], "Text"], Cell[TextData[{ "After determining the correct contour, the next step is to calculate \ residues of the integrand. Since the integrand contains only Gamma \ functions, this task is more or less formal. We don't even need to calculate \ residues of the integrand, but go straightforwardly to the generalized \ hypergeometric functions. The only obstacle is the logarithmic case, which \ occurs when the integrand has multiple poles. We have to separate two \ subclasses here: the integrand that has a finite number of multiple poles, \ and the integrand that has infinitely many multiple poles. The ", StyleBox["Mathematica ", FontSlant->"Italic"], "integration routine has a full implementation of the former case. In the \ latter, the integration artificially restricted by the second order poles, \ since for the higher order poles it would lead to infinite sums with higher \ order polygamma functions. This class of infinite sums are extremely hard to \ deal with, symbolically and numerically. If such a situation is detected ", StyleBox["Integrate", FontFamily->"Courier", FontWeight->"Bold"], " returns the Meijer G-function. However, there are some very special \ transformations of the G-function, which could avoid bulky infinite sums with \ polygamma functions, and give a nice result in terms of known functions. In \ [4] I demonstrated a few transformations that reduce the order of the \ G-function and make it possible to handle special class of Bessel integrals \ in terms of Bessel functions." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["An Example", "Subsection"], Cell["Consider the integral", "Text"], Cell[BoxData[ \(TraditionalForm \`\[CapitalOmega] = \[Integral]\_0\%\[Infinity]\(\(\ sin(x)\)\/\(x\ \((x\^2 + 1)\)\)\) \[DifferentialD]x\)], "Text", TextAlignment->Center, TextJustification->0], Cell["According to the formula (1), we have", "Text"], Cell[BoxData[ \(TraditionalForm \`\(\[CapitalOmega] = \(\[Integral]\_0\%\[Infinity]\( 1\/\(x\^2 + 1\)\) \(sin(1\/\(1/x\))\)\ \[DifferentialD]x\/x = \(1\/\(2\ \[Pi]\ \[ImaginaryI]\)\) \(\[Integral]\+\(\[Gamma] - \[ImaginaryI]\[Infinity]\)\%\(\[Gamma] + \[ImaginaryI]\[Infinity]\)\(\(f\_1\%*\)(s)\)\ \(\(f\_2\%*\)(s)\) \[DifferentialD]s\)\)\ \ \ \)\)], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\(\(f\_1\%*\)(s)\)\ and\ \(\(f\_2\%*\)(s)\)\)]], " are Mellin transforms of ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(-x\)\ \ and\ \(sin(1\/x)\)\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f\_1\^*\)[s] = \[Integral]\_0\%\[Infinity]\( x\^\(s - 1\)\/\(x\^2 + 1\)\) \[DifferentialD]x\)], "Input", CellLabel->"In[15]:="], Cell[BoxData[ FormBox[ RowBox[{"If", "(", RowBox[{ \(Re(s) > 0 \[And] Re(s) < 2\), ",", \(1\/2\ \[Pi]\ \(csc(\(\[Pi]\ s\)\/2)\)\), ",", RowBox[{ SubsuperscriptBox["\[Integral]", "0", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(\(x\^\(s - 1\)\/\(x\^2 + 1\)\) \[DifferentialD]x\)}]}], ")"}], TraditionalForm]], "Output", CellLabel->"Out[15]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(f\_2\^*\)[s] = \[Integral]\_0\%\[Infinity]\( Sin[1\/x]\ x\^\(s - 1\)\) \[DifferentialD]x\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ FormBox[ RowBox[{"If", "(", RowBox[{ \(Re(s) > \(-1\) \[And] Re(s) < 1\), ",", \(\(-\(\[CapitalGamma](\(-s\))\)\)\ \(sin(\(\[Pi]\ s\)\/2)\)\), ",", RowBox[{ SubsuperscriptBox["\[Integral]", "0", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(\(x\^\(s - 1\)\ \(sin(1\/x)\)\) \[DifferentialD]x\)}]}], ")"}], TraditionalForm]], "Output", CellLabel->"Out[16]="] }, Open ]], Cell["Therefore, ", "Text"], Cell[BoxData[ \(TraditionalForm \`\(\[CapitalOmega] = \(-\(\[Pi]\/\(4\ \[Pi]\ \[ImaginaryI]\)\)\) \(\[Integral]\+\(\[Gamma] - \[ImaginaryI]\[Infinity]\)\%\(\[Gamma] + \[ImaginaryI]\[Infinity]\)\(\[CapitalGamma](\(-s\))\) \[DifferentialD]s\)\ \ \ \)\)], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[ "where \[Gamma] is defined by conditions of the existence of Mellin \ transforms"], "Text"], Cell[BoxData[ \(TraditionalForm\`0 < \[Gamma] = Re(s) < 1\)], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "As it follows from the previous section we can transform the integration \ contour ", Cell[BoxData[ \(TraditionalForm \`\((\[Gamma] - \[Infinity], \[Gamma] + \[ImaginaryI]\[Infinity]) \)\)]], " into the right loop ", Cell[BoxData[ \(TraditionalForm\`\(\[ScriptCapitalL]\_\(\ \(+\[Infinity]\)\)\ \)\)]], ". Then, using the residue theorem we evaluate the integral as a sum of \ residues at simple poles ", Cell[BoxData[ \(TraditionalForm\`s = 1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`2\)]], ", ... ." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalOmega] = \(-\(\[Pi]\/2\)\)\ \ \(\[Sum]\+\(k = 1\)\%\[Infinity]\((\(-1\))\)\^k\/\(k!\)\)\)], "Input",\ CellLabel->"In[17]:="], Cell[BoxData[ \(TraditionalForm \`\(-\(\(\((1 - \[ExponentialE])\)\ \[Pi]\)\/\(2\ \[ExponentialE]\)\)\)\)], "Output", CellLabel->"Out[17]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Meijer G-Function", "Subsection"], Cell["\<\ We did not intend to give a complete picture of the Meijer \ G-function here, but only necessary facts. In Version 3.0 the G-function is \ defined by\ \>", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ \(MeijerG[{{a\_1, a\_2, \ ...\ , a\_n}, {a\_\(n + 1\), a\_\(n + 2\), \ ...\ , a\_p}}, {{b\_1, b\_2, \ ...\ , b\_m}, {b\_\(m + 1\), b\_\(m + 2\), \ ...\ , b\_q}}\ , z]\), " ", "=", "\n", "\t ", RowBox[{ RowBox[{\(G\_\(p, q\)\%\(m, n\)\), "(", RowBox[{"z", "\[VerticalSeparator]", RowBox[{GridBox[{ {\(a\_1, \), \(a\_2, \)}, {\(b\_1, \), \(b\_2, \)} }], GridBox[{ {\(...\ , \), \(a\_p\)}, {\(...\ , \), \(b\_q\)} }]}]}], ")"}], " ", "=", "\n", "\t ", \(\(1\/\(2\ \[Pi]\ \[ImaginaryI]\)\) \(\[ContourIntegral]\+\[ScriptCapitalL]\(\(\[Product]\+\(i = 1 \)\%m\( \[CapitalGamma](b\_i + s)\)\ \(\[Product]\+\(i = 1\)\%n \[CapitalGamma]( 1 - a\_i - \ s)\)\ \ \)\/\(\[Product]\+\(i = n + 1\)\%p \( \[CapitalGamma](a\_i + s)\)\ \(\[Product]\+\(i = m + 1\)\%q \[CapitalGamma]( 1 - b\_i - s)\)\)\) \(z\^\(-s\)\) \[DifferentialD]s\)\)}]}], " "}], TraditionalForm]], "Text",\ TextAlignment->Center, TextJustification->0], Cell[TextData[{ "and the contour \[ScriptCapitalL] is a left (or right) loop ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\_\(\(\ \)\(-\[Infinity]\)\)\)]], " (or ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\_\(\(\ \)\(+\[Infinity]\)\)\)]], ") separated poles of ", Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma]\ \((b\_j + s)\)\)]], " from ", Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma]\ \((1 - a\_j - s)\)\)]], ". The current implementation of the Meijer function has two noticeable \ features that differ it from the classical G-function. First, you are not \ allowed to choose or move the contour of integration, and, second, ", StyleBox["MeijerG", FontFamily->"Courier", FontWeight->"Bold"], " has an optional parameter (the classical G-function does not) that \ regulates the branch cut. The Meijer function is supported symbolically and \ numerically." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(MeijerG[{{0}, {}}, {{0}, {}}, z]\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \(TraditionalForm\`1\/\(z + 1\)\)], "Output", CellLabel->"Out[18]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MeijerG[{{\(-1\)}, {1, 2}}, {{0, 1\/2}, {}}, 1\/2]\)], "Input", CellLabel->"In[19]:="], Cell[BoxData[ \(TraditionalForm\`MeijerG({{\(-1\)}, {1, 2}}, {{0, 1\/2}, {}}, 1\/2)\)], "Output", CellLabel->"Out[19]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[%]\)], "Input", CellLabel->"In[20]:="], Cell[BoxData[ \(TraditionalForm\`0.209893208170013956`\)], "Output", CellLabel->"Out[20]="] }, Open ]], Cell[TextData[{ "Only in very trivial cases", StyleBox[" ", FontWeight->"Bold"], StyleBox["MeijerG", FontFamily->"Courier", FontWeight->"Bold"], " is simplified automatically to the lower level special functions. Beyond \ that all further transformations are assigned to ", StyleBox["FunctionExpand", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FunctionExpand[%%]\)], "Input", CellLabel->"In[21]:="], Cell[BoxData[ \(TraditionalForm \`\@\[Pi] - \(\@\[Pi]\ \(\(I\_0\)(1)\)\)\/\[ExponentialE] - \(2\ \@\[Pi]\ \(\(I\_1\)(1)\)\)\/\[ExponentialE]\)], "Output", CellLabel->"Out[21]="] }, Open ]], Cell[TextData[{ StyleBox["MeijerG", FontFamily->"Courier", FontWeight->"Bold"], " is interlaced with indefinite and definite integration, and with solving \ linear differential equations with polynomial coefficients. Here is an \ example related to definite integration:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_1\%\[Infinity]\(\( x\^2\ BesselK[0, x]\)\/\@\(x\^2 - 1\)\) \[DifferentialD]x\)], "Input", CellLabel->"In[22]:="], Cell[BoxData[ \(TraditionalForm \`1\/4\ \@\[Pi]\ \(MeijerG({{}, {\(-\(1\/2\)\)}}, {{\(-1\), 0, 0}, {}}, 1\/4)\)\)], "Output", CellLabel->"Out[22]="] }, Open ]], Cell[TextData[{ "The first element of the second argument of ", StyleBox["MeijerG", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", FontWeight->"Bold"], "which is {-1,0,0}, indicates that the integrand of the correspondent \ contour integral contains the product of three Gamma functions ", Cell[BoxData[ \(TraditionalForm \`\(\[CapitalGamma](s - 1)\) \(\[CapitalGamma](s)\) \(\[CapitalGamma](s)\)\)]], " and has an infinite number of triple poles at ", StyleBox["s", FontSlant->"Italic"], "=-", StyleBox["k", FontSlant->"Italic"], ", ", StyleBox["k", FontSlant->"Italic"], "=0,1,2, ... . From the design point of view it is definitely an advantage \ for ", StyleBox["Integrate", FontFamily->"Courier", FontWeight->"Bold"], " to return a short object, ", StyleBox["MeijerG", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[",", FontWeight->"Bold"], " rather than enormous infinite sums involving derivatives of the Gamma \ function." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Hypergeometric Functions", "Subsection"], Cell["\<\ The essential part of integration is the generalized hypergeometric \ function, which is defined by\ \>", "Text"], Cell[BoxData[ FormBox[ RowBox[{ \(HypergeometricPFQ[{a\_1, ..., a\_p}, {b\_1, ..., b\_q}, z]\), "=", RowBox[{ RowBox[{ RowBox[{\(\[InvisiblePrefixScriptBase]\_p\), SubscriptBox[ StyleBox["F", FontFamily->"Courier", FontSize->14, FontSlant->"Plain", FontTracking->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], "q"]}], "(", RowBox[{"z", "\[VerticalSeparator]", RowBox[{GridBox[{ {\(a\_1, \), \(a\_2, \)}, {\(b\_1, \), \(b\_2, \)} }], GridBox[{ {\(...\ , \), \(a\_p\)}, {\(...\ , \), \(b\_q\)} }]}]}], ")"}], " ", "=", "\n", "\t\t", \(\[Sum]\+\(k = 0 \)\%\[Infinity]\(\(\[Product]\+\(j = 1\)\%p\((a\_j)\)\_k\ \)\/\(\[Product]\+\(j = 1\)\%q\((b\_j)\)\_k\ \)\) z\^k\/\(k!\)\)}]}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\((a)\)\_k\)]], " is a Pochhammer symbol" }], "Text"], Cell[BoxData[ \(TraditionalForm \`\((a\_j)\)\_k\ = \(\[Product]\+\(l = 1\)\%k\((a\_j\ + \ l - 1)\) = \(\[CapitalGamma](a\_j + k)\)\/\(\[CapitalGamma](a\_j)\)\)\)], "Text",\ TextAlignment->Center, TextJustification->0], Cell[TextData[{ "Conditions of convergence can be easily obtained by applying the \ d'Alembert test. It follows, ", Cell[BoxData[ FormBox[ RowBox[{\(\[InvisiblePrefixScriptBase]\_p\), SubscriptBox[ StyleBox["F", FontFamily->"Courier", FontSize->14, FontSlant->"Plain", FontTracking->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], "q"]}], TraditionalForm]]], " converges for all finite ", StyleBox["z", FontSlant->"Italic"], " if ", Cell[BoxData[ \(TraditionalForm\`p \[LessEqual] q\)]], ", and for ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]z\[RightBracketingBar] < 1\)]], " if ", Cell[BoxData[ \(TraditionalForm\`p = q + 1\)]], ". Additional conditions are required on the circle of convergence ", StyleBox["\[LeftBracketingBar]", FontFamily->"Times", FontSize->12, FontWeight->"Roman", FontSlant->"Italic", FontTracking->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["z", FontSlant->"Italic"], StyleBox["\[RightBracketingBar]", FontFamily->"Times", FontSize->12, FontWeight->"Roman", FontSlant->"Italic", FontTracking->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["=1", FontSlant->"Italic"], ". 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Such understanding of divergent integrals is called the principal-value \ or the Cauchy principal-value. It is easy to see that if an integral exists \ in the Riemann sense, it exists in the Cauchy sense. Thus, the class of \ Cauchy integrals is larger than the class of Riemann integrals." }], "Text", CellMargins->{{Inherited, 45}, {Inherited, Inherited}}], Cell[TextData[{ "In Version 3.0, definite integrals in the Riemann sense and \ principal-value integrals are separated by the new option ", StyleBox["PrincipalValue", FontFamily->"Courier", FontWeight->"Bold"], ". If you want to evaluate an integral in the Cauchy sense, set the option \ ", StyleBox["PrincipalValue", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["True", FontFamily->"Courier", FontWeight->"Bold"], " (the default setting is ", StyleBox["False", FontFamily->"Courier", FontWeight->"Bold"], "). 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In this example the restrictions ", Cell[BoxData[ \(TraditionalForm\`\(Re(\[Alpha]) > 0\ \ \)\)]], "and ", Cell[BoxData[ \(TraditionalForm\`Re(\[Lambda]) > 0\)]], " came from conditions of the convergence. Even when a definite integral \ is convergent, some other conditions on parameters might appear. For \ instance, the presence of singularities on the integration path could lead to \ essential changes when the parameters vary. The next section is devoted to \ the convergence of definite integrals. " }], "Text", CellMargins->{{Inherited, 45}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["Convergence ", "Subsection", CellMargins->{{Inherited, 45}, {Inherited, Inherited}}], Cell[TextData[{ "The new integration code contains criteria for the convergence of definite \ integrals. 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" }], "Text", CellMargins->{{Inherited, 45}, {Inherited, Inherited}}], Cell[TextData[{ "If you are sure that a particular integral is convergent or you don't care \ about the convergence, you can avoid testing the convergence by setting the \ option ", StyleBox["GenerateConditions", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["False", FontFamily->"Courier", FontWeight->"Bold"], ". It will make ", StyleBox["Integrate", FontFamily->"Courier", FontWeight->"Bold"], " return an answer a bit faster. " }], "Text", CellMargins->{{Inherited, 45}, {Inherited, Inherited}}], Cell[TextData[{ "Setting ", StyleBox["GenerateConditions", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["False", FontFamily->"Courier", FontWeight->"Bold"], " also lets you evaluate divergent integrals" }], "Text", CellMargins->{{Inherited, 45}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[1\/x, {x, 0, 2}, GenerateConditions \[Rule] False]\)], "Input",\ CellLabel->"In[55]:=", CellMargins->{{Inherited, 45}, {Inherited, Inherited}}], Cell[BoxData[ \(TraditionalForm\`log(2)\)], "Output", CellLabel->"Out[55]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Assumptions", "Subsection", CellMargins->{{Inherited, 45}, {Inherited, Inherited}}], Cell[TextData[{ "The new option ", StyleBox["Assumption", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" ", FontFamily->"Courier"], "is used to specify particular assumptions on parameters in definite \ integrals. Consider the integral with the arbitrary parameter ", Cell[BoxData[ FormBox[ StyleBox["y", FontWeight->"Roman", FontSlant->"Italic"], TraditionalForm]]] }], "Text", CellMargins->{{Inherited, 45}, {Inherited, Inherited}}], Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0 \%\[Infinity]\( \[ExponentialE]\^\(\(-x\^2\) - x\ y\)\/\@x\) \[DifferentialD]x\)], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "Here we set the option ", StyleBox["Assumptions", FontFamily->"Courier", FontWeight->"Bold"], " to ", Cell[BoxData[ \(TraditionalForm\`Re(y) > 0\)]] }], "Text", CellMargins->{{Inherited, 45}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[Exp[\(-x\^2\) - x\ y]\/\@x, {x, 0, \[Infinity]}, Assumptions \[Rule] Re[y] > 0]\)], "Input", CellLabel->"In[56]:=", CellMargins->{{Inherited, 45}, {Inherited, Inherited}}], Cell[BoxData[ \(TraditionalForm \`1\/2\ \[ExponentialE]\^\(y\^2\/8\)\ \@y\ \(\(K\_\(1\/4\)\)(y\^2\/8)\)\)], "Output", 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