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/06.33.21.0052.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2001-10-29