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; }

 Cosh

 http://functions.wolfram.com/01.20.21.1157.01

 Input Form

 Integrate[z^n Cos[d z]^m Cosh[c Sqrt[z] + f z], z] == (2^(-2 - m - 2 n) Binomial[m, m/2] (1 - Mod[m, 2]) (E^(c^2/(4 f)) Sum[(-1)^(-h + j) 4^j (-c)^(-h - j + 2 n) (-c - 2 f Sqrt[z])^(h + j) ((-c - 2 f Sqrt[z])^2/f)^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-c) (-c - 2 f Sqrt[z]) Gamma[(1/2) (1 + h + j), (-c - 2 f Sqrt[z])^2/(4 f)] - 2 f Sqrt[(-c - 2 f Sqrt[z])^2/f] Gamma[(1/2) (2 + h + j), (-c - 2 f Sqrt[z])^2/(4 f)]), {j, 0, n}, {h, 0, j}] + Sum[(-1)^(-h + j) 4^j c^(-h - j + 2 n) (c + 2 f Sqrt[z])^(h + j) (-((c + 2 f Sqrt[z])^2/f))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (c (c + 2 f Sqrt[z]) Gamma[(1/2) (1 + h + j), -((c + 2 f Sqrt[z])^2/(4 f))] + 2 f Sqrt[-((c + 2 f Sqrt[z])^2/f)] Gamma[(1/2) (2 + h + j), -((c + 2 f Sqrt[z])^2/(4 f))]), {j, 0, n}, {h, 0, j}]/ E^(c^2/(4 f))))/f^(2 (1 + n)) + 2^(-2 - m - 2 n) Sum[Binomial[m, k] (Sum[(-1)^(-h + j) 4^j (-c)^(-h - j + 2 n) (-c + 2 (-f - I d (-2 k + m)) Sqrt[z])^(h + j) (-((-c + 2 (-f - I d (-2 k + m)) Sqrt[z])^2/(-f - I d (-2 k + m))))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-c) (-c + 2 (-f - I d (-2 k + m)) Sqrt[z]) Gamma[(1/2) (1 + h + j), -((-c + 2 (-f - I d (-2 k + m)) Sqrt[z])^ 2/(4 (-f - I d (-2 k + m))))] + 2 (-f - I d (-2 k + m)) Sqrt[-((-c + 2 (-f - I d (-2 k + m)) Sqrt[z])^2/(-f - I d (-2 k + m)))] Gamma[(1/2) (2 + h + j), -((-c + 2 (-f - I d (-2 k + m)) Sqrt[z])^2/(4 (-f - I d (-2 k + m))))]), {j, 0, n}, {h, 0, j}]/ (E^(c^2/(4 (-f - I d (-2 k + m)))) (-f - I d (-2 k + m))^ (2 (1 + n))) + Sum[(-1)^(-h + j) 4^j c^(-h - j + 2 n) (c + 2 (f - I d (-2 k + m)) Sqrt[z])^(h + j) (-((c + 2 (f - I d (-2 k + m)) Sqrt[z])^2/(f - I d (-2 k + m))))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (c (c + 2 (f - I d (-2 k + m)) Sqrt[z]) Gamma[(1/2) (1 + h + j), -((c + 2 (f - I d (-2 k + m)) Sqrt[z])^2/(4 (f - I d (-2 k + m))))] + 2 (f - I d (-2 k + m)) Sqrt[-((c + 2 (f - I d (-2 k + m)) Sqrt[z])^2/(f - I d (-2 k + m)))] Gamma[(1/2) (2 + h + j), -((c + 2 (f - I d (-2 k + m)) Sqrt[z])^2/(4 (f - I d (-2 k + m))))]), {j, 0, n}, {h, 0, j}]/ (E^(c^2/(4 (f - I d (-2 k + m)))) (f - I d (-2 k + m))^(2 (1 + n))) + Sum[(-1)^(-h + j) 4^j (-c)^(-h - j + 2 n) (-c + 2 (-f + I d (-2 k + m)) Sqrt[z])^(h + j) (-((-c + 2 (-f + I d (-2 k + m)) Sqrt[z])^2/(-f + I d (-2 k + m))))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-c) (-c + 2 (-f + I d (-2 k + m)) Sqrt[z]) Gamma[(1/2) (1 + h + j), -((-c + 2 (-f + I d (-2 k + m)) Sqrt[z])^ 2/(4 (-f + I d (-2 k + m))))] + 2 (-f + I d (-2 k + m)) Sqrt[-((-c + 2 (-f + I d (-2 k + m)) Sqrt[z])^2/(-f + I d (-2 k + m)))] Gamma[(1/2) (2 + h + j), -((-c + 2 (-f + I d (-2 k + m)) Sqrt[z])^2/(4 (-f + I d (-2 k + m))))]), {j, 0, n}, {h, 0, j}]/ (E^(c^2/(4 (-f + I d (-2 k + m)))) (-f + I d (-2 k + m))^ (2 (1 + n))) + Sum[(-1)^(-h + j) 4^j c^(-h - j + 2 n) (c + 2 (f + I d (-2 k + m)) Sqrt[z])^(h + j) (-((c + 2 (f + I d (-2 k + m)) Sqrt[z])^2/(f + I d (-2 k + m))))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (c (c + 2 (f + I d (-2 k + m)) Sqrt[z]) Gamma[(1/2) (1 + h + j), -((c + 2 (f + I d (-2 k + m)) Sqrt[z])^2/(4 (f + I d (-2 k + m))))] + 2 (f + I d (-2 k + m)) Sqrt[-((c + 2 (f + I d (-2 k + m)) Sqrt[z])^2/(f + I d (-2 k + m)))] Gamma[(1/2) (2 + h + j), -((c + 2 (f + I d (-2 k + m)) Sqrt[z])^2/(4 (f + I d (-2 k + m))))]), {j, 0, n}, {h, 0, j}]/ (E^(c^2/(4 (f + I d (-2 k + m)))) (f + I d (-2 k + m))^(2 (1 + n)))), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 z n cos m ( d z ) cosh ( z c + f z ) z 2 - m - 2 n - 2 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( c 2 4 f j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - c ) - h - j + 2 n ( - c - 2 f z ) h + j ( ( - c - 2 f z ) 2 f ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - c ( - c - 2 f z ) Γ ( 1 2 ( h + j + 1 ) , ( - c - 2 f z ) 2 4 f ) - 2 f ( - c - 2 f z ) 2 f Γ ( 1 2 ( h + j + 2 ) , ( - c - 2 f z ) 2 4 f ) ) + - c 2 4 f j = 0 n h = 0 j ( - 1 ) j - h 4 j c - h - j + 2 n ( c + 2 f z ) h + j ( - ( c + 2 f z ) 2 f ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( c + 2 f z ) Γ ( 1 2 ( h + j + 1 ) , - ( c + 2 f z ) 2 4 f ) + 2 - ( c + 2 f z ) 2 f f Γ ( 1 2 ( h + j + 2 ) , - ( c + 2 f z ) 2 4 f ) ) ) f - 2 ( n + 1 ) + 2 - m - 2 n - 2 k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - c 2 4 ( d ( m - 2 k ) - f ) ( j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - c ) - h - j + 2 n ( 2 ( d ( m - 2 k ) - f ) z - c ) h + j ( - ( 2 ( d ( m - 2 k ) - f ) z - c ) 2 d ( m - 2 k ) - f ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 ( d ( m - 2 k ) - f ) - ( 2 ( d ( m - 2 k ) - f ) z - c ) 2 d ( m - 2 k ) - f Γ ( 1 2 ( h + j + 2 ) , - ( 2 ( d ( m - 2 k ) - f ) z - c ) 2 4 ( d ( m - 2 k ) - f ) ) - c ( 2 ( d ( m - 2 k ) - f ) z - c ) Γ ( 1 2 ( h + j + 1 ) , - ( 2 ( d ( m - 2 k ) - f ) z - c ) 2 4 ( d ( m - 2 k ) - f ) ) ) ) ( d ( m - 2 k ) - f ) - 2 ( n + 1 ) + - c 2 4 ( f + d ( m - 2 k ) ) ( f + d ( m - 2 k ) ) - 2 ( n + 1 ) j = 0 n h = 0 j ( - 1 ) j - h 4 j c - h - j + 2 n ( c + 2 ( f + d ( m - 2 k ) ) z ) h + j ( - ( c + 2 ( f + d ( m - 2 k ) ) z ) 2 f + d ( m - 2 k ) ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( c + 2 ( f + d ( m - 2 k ) ) z ) Γ ( 1 2 ( h + j + 1 ) , - ( c + 2 ( f + d ( m - 2 k ) ) z ) 2 4 ( f + d ( m - 2 k ) ) ) + 2 - ( c + 2 ( f + d ( m - 2 k ) ) z ) 2 f + d ( m - 2 k ) ( f + d ( m - 2 k ) ) Γ ( 1 2 ( h + j + 2 ) , - ( c + 2 ( f + d ( m - 2 k ) ) z ) 2 4 ( f + d ( m - 2 k ) ) ) ) + - c 2 4 ( - f - d ( m - 2 k ) ) ( - f - d ( m - 2 k ) ) - 2 ( n + 1 ) j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - c ) - h - j + 2 n ( 2 ( - f - d ( m - 2 k ) ) z - c ) h + j ( - ( 2 ( - f - d ( m - 2 k ) ) z - c ) 2 - f - d ( m - 2 k ) ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 ( - f - d ( m - 2 k ) ) - ( 2 ( - f - d ( m - 2 k ) ) z - c ) 2 - f - d ( m - 2 k ) Γ ( 1 2 ( h + j + 2 ) , - ( 2 ( - f - d ( m - 2 k ) ) z - c ) 2 4 ( - f - d ( m - 2 k ) ) ) - c ( 2 ( - f - d ( m - 2 k ) ) z - c ) Γ ( 1 2 ( h + j + 1 ) , - ( 2 ( - f - d ( m - 2 k ) ) z - c ) 2 4 ( - f - d ( m - 2 k ) ) ) ) + - c 2 4 ( f - d ( m - 2 k ) ) ( f - d ( m - 2 k ) ) - 2 ( n + 1 ) j = 0 n h = 0 j ( - 1 ) j - h 4 j c - h - j + 2 n ( c + 2 ( f - d ( m - 2 k ) ) z ) h + j ( - ( c + 2 ( f - d ( m - 2 k ) ) z ) 2 f - d ( m - 2 k ) ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( c + 2 ( f - d ( m - 2 k ) ) z ) Γ ( 1 2 ( h + j + 1 ) , - ( c + 2 ( f - d ( m - 2 k ) ) z ) 2 4 ( f - d ( m - 2 k ) ) ) + 2 - ( c + 2 ( f - d ( m - 2 k ) ) z ) 2 f - d ( m - 2 k ) ( f - d ( m - 2 k ) ) Γ ( 1 2 ( h + j + 2 ) , - ( c + 2 ( f - d ( m - 2 k ) ) z ) 2 4 ( f - d ( m - 2 k ) ) ) ) ) /; n m + Condition z z n d z m z 1 2 c f z 2 -1 m -1 2 n -2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 c 2 4 f -1 h 0 j j 0 n -1 j -1 h 4 j -1 c -1 h -1 j 2 n -1 c -1 2 f z 1 2 h j -1 c -1 2 f z 1 2 2 f -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j -1 c -1 c -1 2 f z 1 2 Gamma 1 2 h j 1 -1 c -1 2 f z 1 2 2 4 f -1 -1 2 f -1 c -1 2 f z 1 2 2 f -1 1 2 Gamma 1 2 h j 2 -1 c -1 2 f z 1 2 2 4 f -1 -1 c 2 4 f -1 h 0 j j 0 n -1 j -1 h 4 j c -1 h -1 j 2 n c 2 f z 1 2 h j -1 c 2 f z 1 2 2 f -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c c 2 f z 1 2 Gamma 1 2 h j 1 -1 c 2 f z 1 2 2 4 f -1 2 -1 c 2 f z 1 2 2 f -1 1 2 f Gamma 1 2 h j 2 -1 c 2 f z 1 2 2 4 f -1 f -2 n 1 2 -1 m -1 2 n -2 k 0 m -1 2 -1 Binomial m k -1 c 2 4 d m -1 2 k -1 f -1 h 0 j j 0 n -1 j -1 h 4 j -1 c -1 h -1 j 2 n 2 d m -1 2 k -1 f z 1 2 -1 c h j -1 2 d m -1 2 k -1 f z 1 2 -1 c 2 d m -1 2 k -1 f -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j 2 d m -1 2 k -1 f -1 2 d m -1 2 k -1 f z 1 2 -1 c 2 d m -1 2 k -1 f -1 1 2 Gamma 1 2 h j 2 -1 2 d m -1 2 k -1 f z 1 2 -1 c 2 4 d m -1 2 k -1 f -1 -1 c 2 d m -1 2 k -1 f z 1 2 -1 c Gamma 1 2 h j 1 -1 2 d m -1 2 k -1 f z 1 2 -1 c 2 4 d m -1 2 k -1 f -1 d m -1 2 k -1 f -2 n 1 -1 c 2 4 f d m -1 2 k -1 f d m -1 2 k -2 n 1 h 0 j j 0 n -1 j -1 h 4 j c -1 h -1 j 2 n c 2 f d m -1 2 k z 1 2 h j -1 c 2 f d m -1 2 k z 1 2 2 f d m -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c c 2 f d m -1 2 k z 1 2 Gamma 1 2 h j 1 -1 c 2 f d m -1 2 k z 1 2 2 4 f d m -1 2 k -1 2 -1 c 2 f d m -1 2 k z 1 2 2 f d m -1 2 k -1 1 2 f d m -1 2 k Gamma 1 2 h j 2 -1 c 2 f d m -1 2 k z 1 2 2 4 f d m -1 2 k -1 -1 c 2 4 -1 f -1 d m -1 2 k -1 -1 f -1 d m -1 2 k -2 n 1 h 0 j j 0 n -1 j -1 h 4 j -1 c -1 h -1 j 2 n 2 -1 f -1 d m -1 2 k z 1 2 -1 c h j -1 2 -1 f -1 d m -1 2 k z 1 2 -1 c 2 -1 f -1 d m -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j 2 -1 f -1 d m -1 2 k -1 2 -1 f -1 d m -1 2 k z 1 2 -1 c 2 -1 f -1 d m -1 2 k -1 1 2 Gamma 1 2 h j 2 -1 2 -1 f -1 d m -1 2 k z 1 2 -1 c 2 4 -1 f -1 d m -1 2 k -1 -1 c 2 -1 f -1 d m -1 2 k z 1 2 -1 c Gamma 1 2 h j 1 -1 2 -1 f -1 d m -1 2 k z 1 2 -1 c 2 4 -1 f -1 d m -1 2 k -1 -1 c 2 4 f -1 d m -1 2 k -1 f -1 d m -1 2 k -2 n 1 h 0 j j 0 n -1 j -1 h 4 j c -1 h -1 j 2 n c 2 f -1 d m -1 2 k z 1 2 h j -1 c 2 f -1 d m -1 2 k z 1 2 2 f -1 d m -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c c 2 f -1 d m -1 2 k z 1 2 Gamma 1 2 h j 1 -1 c 2 f -1 d m -1 2 k z 1 2 2 4 f -1 d m -1 2 k -1 2 -1 c 2 f -1 d m -1 2 k z 1 2 2 f -1 d m -1 2 k -1 1 2 f -1 d m -1 2 k Gamma 1 2 h j 2 -1 c 2 f -1 d m -1 2 k z 1 2 2 4 f -1 d m -1 2 k -1 n m SuperPlus [/itex]

 Rule Form

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

 2002-12-18