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 FresnelC

 http://functions.wolfram.com/06.33.21.0015.01

 Input Form

 Integrate[z^n E^(b z) FresnelC[a z], z] == (-(-b)^(-1 - n)) FresnelC[a z] Gamma[1 + n, (-b) z] + a (-b)^(-1 - n) n! Sum[(1/m!) (-b)^m (((-((-I) a^2)^(-1 - m)) Sum[(2^((1/2) (-3 + k)) (-b)^(-k + m) Pi^((1/2) (-1 + k - 2 m)) (b - I a^2 Pi z)^(1 + k) Binomial[m, k] Gamma[(1 + k)/2, (I (I b + a^2 Pi z)^2)/(2 a^2 Pi)])/ ((I (I b + a^2 Pi z)^2)/a^2)^((k + 1)/2), {k, 0, m}])/ E^((I b^2)/(2 a^2 Pi)) - (I a^2)^(-1 - m) E^((I b^2)/(2 a^2 Pi)) Sum[(2^((1/2) (-3 + k)) (-b)^(-k + m) Pi^((1/2) (-1 + k - 2 m)) (b + I a^2 Pi z)^(1 + k) Binomial[m, k] Gamma[(1 + k)/2, (I (b + I a^2 Pi z)^2)/(2 a^2 Pi)])/((I (b + I a^2 Pi z)^2)/a^2)^ ((k + 1)/2), {k, 0, m}]), {m, 0, n}] /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n b z C TagBox["C", FresnelC] ( a z ) z a ( - b ) - n - 1 n ! m = 0 n 1 m ! ( - b ) m ( - ( - a 2 ) - m - 1 - b 2 2 a 2 π k = 0 m 2 k - 3 2 ( - b ) m - k π 1 2 ( k - 2 m - 1 ) ( b - a 2 π z ) k + 1 ( ( π z a 2 + b ) 2 a 2 ) - 1 2 ( k + 1 ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( k + 1 2 , ( π z a 2 + b ) 2 2 a 2 π ) - ( a 2 ) - m - 1 b 2 2 a 2 π k = 0 m 2 k - 3 2 ( - b ) m - k π 1 2 ( k - 2 m - 1 ) ( π z a 2 + b ) k + 1 ( ( π z a 2 + b ) 2 a 2 ) - 1 2 ( k + 1 ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( k + 1 2 , ( π z a 2 + b ) 2 2 a 2 π ) ) - ( - b ) - n - 1 C TagBox["C", FresnelC] ( a z ) Γ ( n + 1 , - b z ) /; n Condition z z n b z FresnelC a z a -1 b -1 n -1 n m 0 n 1 m -1 -1 b m -1 -1 a 2 -1 m -1 -1 b 2 2 a 2 -1 k 0 m 2 k -3 2 -1 -1 b m -1 k 1 2 k -1 2 m -1 b -1 a 2 z k 1 z a 2 b 2 a 2 -1 -1 1 2 k 1 Binomial m k Gamma k 1 2 -1 z a 2 b 2 2 a 2 -1 -1 a 2 -1 m -1 b 2 2 a 2 -1 k 0 m 2 k -3 2 -1 -1 b m -1 k 1 2 k -1 2 m -1 z a 2 b k 1 z a 2 b 2 a 2 -1 -1 1 2 k 1 Binomial m k Gamma k 1 2 -1 z a 2 b 2 2 a 2 -1 -1 -1 b -1 n -1 FresnelC a z Gamma n 1 -1 b z n [/itex]

 Rule Form

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

 2001-10-29