Hyperbolic sine: Integration (formula 01.19.21.2229)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Integration

Indefinite integration

Involving functions of the direct function and exponential function

Involving products of powers of two direct functions and exponential function

Involving product of power of the direct function, the direct function and algebraic functions of exp

Involving e^{p z}(a+b e^{d z})^beta sinh(e z)sinh^v(c z)

http://functions.wolfram.com/01.19.21.2229.01

Input Form

Integrate[E^(p z) (a + b E^(d z))^\[Beta] Sinh[e z] Sinh[c z]^v, z] == (1/((e - p) (e + p))) (((I/2)^(1 + v) (a + b E^(d z))^\[Beta] Binomial[v, v/2] ((-E^((I Pi)/2 + (-e + p) z)) (e + p) Hypergeometric2F1[(-e + p)/d, -\[Beta], (d - e + p)/d, -((b E^(d z))/a)] + E^(-((I Pi)/2) + (e + p) z) (e - p) Hypergeometric2F1[(e + p)/d, -\[Beta], (d + e + p)/d, -((b E^(d z))/a)]) (1 - Mod[v, 2]))/(1 + (b E^(d z))/a)^\[Beta]) + ((I/2)^(1 + v) (a + b E^(d z))^\[Beta] Sum[(-1)^s Binomial[v, s] (Cos[(1/2) Pi (1 - v)] (((-E^((e + p + 2 c s - c v) z)) (-e + p - 2 c s + c v) Hypergeometric2F1[(e + p + 2 c s - c v)/d, -\[Beta], (d + e + p + 2 c s - c v)/d, -((b E^(d z))/a)] + E^((-e + p - 2 c s + c v) z) (-e - p - 2 c s + c v) Hypergeometric2F1[(-e + p - 2 c s + c v)/d, -\[Beta], (d - e + p - 2 c s + c v)/d, -((b E^(d z))/a)])/ ((-e - p - 2 c s + c v) (-e + p - 2 c s + c v))) + Cos[(1/2) Pi (1 + v)] ((E^((-e + p + 2 c s - c v) z) (-e - p + 2 c s - c v) Hypergeometric2F1[(-e + p + 2 c s - c v)/ d, -\[Beta], (d - e + p + 2 c s - c v)/d, -((b E^(d z))/a)] - E^((e + p - 2 c s + c v) z) (-e + p + 2 c s - c v) Hypergeometric2F1[(e + p - 2 c s + c v)/d, -\[Beta], (d + e + p - 2 c s + c v)/d, -((b E^(d z))/a)])/ ((-e - p + 2 c s - c v) (-e + p + 2 c s - c v))) + I (E^((e + p + 2 c s - c v) z) (-e + p - 2 c s + c v) Hypergeometric2F1[(e + p + 2 c s - c v)/d, -\[Beta], (d + e + p + 2 c s - c v)/d, -((b E^(d z))/a)] + E^((-e + p - 2 c s + c v) z) (-e - p - 2 c s + c v) Hypergeometric2F1[(-e + p - 2 c s + c v)/d, -\[Beta], (d - e + p - 2 c s + c v)/d, -((b E^(d z))/a)]) (Sin[(1/2) Pi (1 - v)]/((-e - p - 2 c s + c v) (-e + p - 2 c s + c v))) + I (E^((-e + p + 2 c s - c v) z) (-e - p + 2 c s - c v) Hypergeometric2F1[(-e + p + 2 c s - c v)/d, -\[Beta], (d - e + p + 2 c s - c v)/d, -((b E^(d z))/a)] + E^((e + p - 2 c s + c v) z) (-e + p + 2 c s - c v) Hypergeometric2F1[(e + p - 2 c s + c v)/d, -\[Beta], (d + e + p - 2 c s + c v)/d, -((b E^(d z))/a)]) (Sin[(1/2) Pi (1 + v)]/((-e - p + 2 c s - c v) (-e + p + 2 c s - c v)))), {s, 0, Floor[(1/2) (-1 + v)]}])/ (1 + (b E^(d z))/a)^\[Beta] /; Element[v, Integers] && v > 0

Standard Form

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MathML Form

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<mi> v </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mi> a </mi> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> β </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> β </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> s </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> v </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> v </mi> </mtd> </mtr> <mtr> <mtd> <mi> s </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mi> d </mi> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mi> β </mi> </mrow> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> d </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mi> d </mi> </mfrac> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mi> a </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List["-", "e"]], "+", "p", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], "d"], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["d", "-", "e", "+", "p", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], "d"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", 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F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> e </mi> <mo> + </mo> <mi> p </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mi> d </mi> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mi> β </mi> </mrow> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> d </mi> <mo> + </mo> <mi> e </mi> <mo> + </mo> <mi> p </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mi> d </mi> </mfrac> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mi> a </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> v </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> + </mo> <mi> p </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> - </mo> <mi> p </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> 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</mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> + </mo> <mi> p </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> e </mi> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mi> d </mi> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mi> β </mi> </mrow> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> d </mi> <mo> + </mo> <mi> e </mi> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mi> d </mi> </mfrac> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mi> a </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["e", "+", "p", "-", RowBox[List["2", " ", "c", " ", 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<mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> - </mo> <mi> p </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> + </mo> <mi> p </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> v </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> + </mo> <mi> p 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</mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> ⅈ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> v </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> β </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mi> a </mi> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> β </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> v </mi> </mtd> </mtr> <mtr> <mtd> <mfrac> <mi> v </mi> <mn> 2 </mn> </mfrac> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> e </mi> <mo> + </mo> <mi> p </mi> </mrow> <mi> d </mi> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mi> β </mi> </mrow> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> d </mi> <mo> + </mo> <mi> e </mi> <mo> + </mo> <mi> p </mi> </mrow> <mi> d </mi> </mfrac> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mi> a </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], 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type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <ci> e </ci> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> s </ci> </apply> </apply> <apply> <times /> <ci> c </ci> <ci> v </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <ci> v </ci> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <ci> e </ci> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> s </ci> </apply> </apply> 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Rule Form

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

2002-12-18