html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 FresnelC

 http://functions.wolfram.com/06.33.21.0078.01

 Input Form

 Integrate[z^n Cosh[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! (E^((I b^2)/(2 a^2 Pi)) Sum[((a^(-2 m - 2) (I b)^m)/m!) ((-E^(-((I b^2)/(a^2 Pi)))) Sum[2^((1/2) (-3 + k)) b^(-k + m) Pi^((1/2) (-1 + k - 2 m)) (I (I b - a^2 Pi z))^(1 + k) ((I (I b - a^2 Pi z)^2)/a^2)^((1/2) (-1 - k)) Binomial[m, k] Gamma[(1 + k)/2, (I (I b - a^2 Pi z)^2)/(2 a^2 Pi)], {k, 0, m}] + (-1)^m Sum[2^((1/2) (-3 + k)) b^(-k + m) Pi^((1/2) (-1 + k - 2 m)) (I (I b + a^2 Pi z))^(1 + k) (-((I (I b + a^2 Pi z)^2)/a^2))^((1/2) (-1 - k)) Binomial[m, k] Gamma[(1 + k)/2, -((I (I b + a^2 Pi z)^2)/(2 a^2 Pi))], {k, 0, m}]), {m, 0, n}] - (-1)^n E^((I b^2)/(2 a^2 Pi)) Sum[(((-1)^m a^(-2 m - 2) (I b)^m)/m!) ((-1)^m Sum[2^((1/2) (-3 + k)) (-b)^(-k + m) Pi^((1/2) (-1 + k - 2 m)) (-((I (I b - a^2 Pi z)^2)/a^2))^ ((1/2) (-1 - k)) (I ((-I) b + 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))], {k, 0, m}] - Sum[2^((1/2) (-3 + k)) (-b)^(-k + m) Pi^((1/2) (-1 + k - 2 m)) ((-I) (I b + a^2 Pi z))^(1 + k) ((I (I b + a^2 Pi z)^2)/a^2)^((1/2) (-1 - k)) Binomial[m, k] Gamma[(1 + k)/2, (I (I b + a^2 Pi z)^2)/(2 a^2 Pi)], {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 cosh ( b z ) C TagBox["C", FresnelC] ( a z ) z 1 2 b - n - 1 ( 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 1 2 b -1 n -1 FresnelC a z -1 n Gamma n 1 -1 b z -1 Gamma n 1 b z a n 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 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 -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 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 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 -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 -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 -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