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 FresnelC

 http://functions.wolfram.com/06.33.21.0070.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2001-10-29