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 FresnelC

 http://functions.wolfram.com/06.33.21.0039.01

 Input Form

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

 Standard Form

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 MathML Form

 z n cos ( b z ) C TagBox["C", FresnelC] ( a z ) z n b - n - 1 2 ( C TagBox["C", FresnelC] ( a z ) ( ( - 1 ) n Γ ( n + 1 , b z ) - Γ ( n + 1 , - b z ) ) - a n ! ( - b 2 2 a 2 π m = 0 n a - 2 m - 2 b m m ! ( ( - 1 ) m 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 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 ( ( b - a 2 π z ) 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 , ( b - a 2 π z ) 2 2 a 2 π ) ) ) - ( - 1 ) n - b 2 2 a 2 π m = 0 n ( - 1 ) m a - 2 m - 2 b m m ! ( ( - 1 ) m k = 0 m 2 k - 3 2 ( b ) m - k π 1 2 ( k - 2 m - 1 ) ( - ( b - a 2 π z ) 2 a 2 ) 1 2 ( - k - 1 ) ( ( a 2 π z - b ) ) 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 , - ( b - a 2 π z ) 2 2 a 2 π ) - b 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 ) ) ) /; n Condition z z n b z FresnelC a z n b -1 n -1 2 -1 FresnelC a z -1 n Gamma n 1 b z -1 Gamma n 1 -1 b z -1 a n -1 b 2 2 a 2 -1 m 0 n a -2 m -2 b m m -1 -1 m 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 -1 z a 2 b 2 a 2 -1 1 2 -1 k -1 Binomial m k Gamma k 1 2 -1 -1 z a 2 b 2 2 a 2 -1 -1 b 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 b -1 a 2 z 2 a 2 -1 1 2 -1 k -1 Binomial m k Gamma k 1 2 -1 b -1 a 2 z 2 2 a 2 -1 -1 -1 n -1 b 2 2 a 2 -1 m 0 n -1 m a -2 m -2 b m m -1 -1 m k 0 m 2 k -3 2 -1 b m -1 k 1 2 k -1 2 m -1 -1 b -1 a 2 z 2 a 2 -1 1 2 -1 k -1 a 2 z -1 b k 1 Binomial m k Gamma k 1 2 -1 -1 b -1 a 2 z 2 2 a 2 -1 -1 b 2 a 2 -1 k 0 m 2 k -3 2 -1 b m -1 k 1 2 k -1 2 m -1 -1 z a 2 b k 1 z a 2 b 2 a 2 -1 1 2 -1 k -1 Binomial m k Gamma k 1 2 -1 z a 2 b 2 2 a 2 -1 n [/itex]

 Rule Form

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

 2001-10-29