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 SinhIntegral

 http://functions.wolfram.com/06.39.21.0064.01

 Input Form

 Integrate[z^n ExpIntegralEi[a z] SinhIntegral[a z], z] == (1/(n + 1)) (z^(1 + n) ExpIntegralEi[a z] + (-a)^(-1 - n) Gamma[1 + n, (-a) z]) SinhIntegral[a z] - (a^(-1 - n)/(2 (1 + n))) (((-1)^n Gamma[1 + n, (-a) z] + Gamma[1 + n, a z]) ExpIntegralEi[a z] - (-1)^n n! ExpIntegralEi[2 a z] - n! Log[z] + (-1)^n n! Sum[Gamma[k, -2 a z]/(2^k k!), {k, 1, n}] - n! Sum[(a z)^k/(k k!), {k, 1, n}]) - (((-a)^(-1 - n) n!)/(2 (1 + n))) (ExpIntegralEi[2 a z] - Log[z] - Sum[(((-a) z)^k/k + Gamma[k, -2 a z]/2^k)/k!, {k, 1, n}]) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n Ei ( a z ) Shi ( a z ) z ( Γ ( n + 1 , - a z ) ( - a ) - n - 1 + z n + 1 Ei ( a z ) ) Shi ( a z ) n + 1 - ( - a ) - n - 1 n ! 2 ( n + 1 ) ( Ei ( 2 a z ) - log ( z ) - k = 1 n 1 k ! ( ( - a z ) k k + 2 - k Γ ( k , - 2 a z ) ) ) - 1 2 ( n + 1 ) ( a - n - 1 ( - ( - 1 ) n Ei ( 2 a z ) n ! - log ( z ) n ! + ( - 1 ) n n ! k = 1 n 2 - k Γ ( k , - 2 a z ) k ! - n ! k = 1 n ( a z ) k k k ! + Ei ( a z ) ( ( - 1 ) n Γ ( n + 1 , - a z ) + Γ ( n + 1 , a z ) ) ) ) /; n Condition z z n ExpIntegralEi a z SinhIntegral a z Gamma n 1 -1 a z -1 a -1 n -1 z n 1 ExpIntegralEi a z SinhIntegral a z n 1 -1 -1 -1 a -1 n -1 n 2 n 1 -1 ExpIntegralEi 2 a z -1 z -1 k 1 n 1 k -1 -1 a z k k -1 2 -1 k Gamma k -2 a z -1 1 2 n 1 -1 a -1 n -1 -1 -1 n ExpIntegralEi 2 a z n -1 z n -1 n n k 1 n 2 -1 k Gamma k -2 a z k -1 -1 n k 1 n a z k k k -1 ExpIntegralEi a z -1 n Gamma n 1 -1 a z Gamma n 1 a z n [/itex]

 Rule Form

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

 2001-10-29