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; }

 InverseGammaRegularized

 http://functions.wolfram.com/06.12.06.0005.01

 Input Form

 InverseGammaRegularized[a, z] \[Proportional] InverseGammaRegularized[a, Subscript[z, 0]] - E^w w^(1 - a) Gamma[a] (z - Subscript[z, 0]) + (1/2) E^(2 w) w^(1 - 2 a) (1 - a + w) Gamma[a]^2 (z - Subscript[z, 0])^2 - (1/6) E^(3 w) w^(1 - 3 a) (1 + 2 a^2 + 2 w (2 + w) - a (3 + 4 w)) Gamma[a]^3 (z - Subscript[z, 0])^3 + (1/24) E^(4 w) w^(1 - 4 a) (1 - 6 a^3 + a^2 (11 + 18 w) + w (11 + 6 w (3 + w)) - a (6 + w (29 + 18 w))) Gamma[a]^4 (z - Subscript[z, 0])^4 - (1/120) E^(5 w) w^(1 - 5 a) (1 + 24 a^4 - 2 a^3 (25 + 48 w) + 2 w (1 + w) (13 + 12 w (3 + w)) + a^2 (35 + 4 w (49 + 36 w)) - 2 a (5 + w (63 + w (121 + 48 w)))) Gamma[a]^5 (z - Subscript[z, 0])^5 + (1/720) E^(6 w) w^(1 - 6 a) (1 - 120 a^5 + a^4 (274 + 600 w) - 3 a^3 (75 + 474 w + 400 w^2) + a^2 (85 + 3 w (399 + 874 w + 400 w^2)) + w (57 + 2 w (212 + w (437 + 60 w (5 + w)))) - a (15 + 2 w (216 + w (923 + w (1037 + 300 w))))) Gamma[a]^6 (z - Subscript[z, 0])^6 - (1/5040) E^(7 w) w^(1 - 7 a) (1 + a (-21 + a (175 + a (-735 + 4 a (406 + 9 a (-49 + 20 a))))) + 120 w - 24 a (54 + a (-229 + 6 a (79 + a (-79 + 30 a)))) w + 6 (-1 + a) (-1 + 2 a) (269 + 36 a (-27 + 25 a)) w^2 - 8 (-1 + a) (755 + 18 a (-129 + 100 a)) w^3 + 36 (-1 + a) (-229 + 300 a) w^4 - 4320 (-1 + a) w^5 + 720 w^6) Gamma[a]^7 (z - Subscript[z, 0])^7 + (1/40320) E^(8 w) w^(1 - 8 a) (1 - 28 a + 322 a^2 - 1960 a^3 + 6769 a^4 - 13132 a^5 + 13068 a^6 - 5040 a^7 + (-1 + a) (-1 + 2 a) (-1 + 3 a) (-247 + a (2097 + 10 a (-599 + 588 a))) w - 6 (-1 + a) (-1 + 2 a) (-947 + a (5891 + 60 a (-206 + 147 a))) w^2 + 2 (-1 + a) (-17729 + 2 a (45001 + 90 a (-853 + 490 a))) w^3 - 20 (-1 + a) (4175 + 9 a (-1343 + 980 a)) w^4 + 36 (-1 + a) (-2323 + 2940 a) w^5 - 35280 (-1 + a) w^6 + 5040 w^7) Gamma[a]^8 (z - Subscript[z, 0])^8 + \[Ellipsis] /; (z -> Subscript[z, 0]) && w = InverseGammaRegularized[a, Subscript[z, 0]]

 Standard Form

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 MathML Form

 Q GammaRegularized - 1 ( a , z ) Q GammaRegularized - 1 ( a , z 0 ) - w Γ ( a ) w 1 - a ( z - z 0 ) + 1 2 2 w ( - a + w + 1 ) Γ ( a ) 2 w 1 - 2 a ( z - z 0 ) 2 - 1 6 3 w ( 2 a 2 - ( 4 w + 3 ) a + 2 w ( w + 2 ) + 1 ) Γ ( a ) 3 w 1 - 3 a ( z - z 0 ) 3 + 1 24 4 w ( - 6 a 3 + ( 18 w + 11 ) a 2 - ( w ( 18 w + 29 ) + 6 ) a + w ( 6 w ( w + 3 ) + 11 ) + 1 ) Γ ( a ) 4 w 1 - 4 a ( z - z 0 ) 4 - 1 120 5 w ( 24 a 4 - 2 ( 48 w + 25 ) a 3 + ( 4 w ( 36 w + 49 ) + 35 ) a 2 - 2 ( w ( w ( 48 w + 121 ) + 63 ) + 5 ) a + 2 w ( w + 1 ) ( 12 w ( w + 3 ) + 13 ) + 1 ) Γ ( a ) 5 w 1 - 5 a ( z - z 0 ) 5 + 1 720 6 w ( - 120 a 5 + ( 600 w + 274 ) a 4 - 3 ( 400 w 2 + 474 w + 75 ) a 3 + ( 3 w ( 400 w 2 + 874 w + 399 ) + 85 ) a 2 - ( 2 w ( w ( w ( 300 w + 1037 ) + 923 ) + 216 ) + 15 ) a + w ( 2 w ( w ( 60 w ( w + 5 ) + 437 ) + 212 ) + 57 ) + 1 ) Γ ( a ) 6 w 1 - 6 a ( z - z 0 ) 6 - 1 5040 7 w ( 720 w 6 - 4320 ( a - 1 ) w 5 + 36 ( a - 1 ) ( 300 a - 229 ) w 4 - 8 ( a - 1 ) ( 18 a ( 100 a - 129 ) + 755 ) w 3 + 6 ( a - 1 ) ( 2 a - 1 ) ( 36 a ( 25 a - 27 ) + 269 ) w 2 - 24 a ( a ( 6 a ( a ( 30 a - 79 ) + 79 ) - 229 ) + 54 ) w + 120 w + a ( a ( a ( 4 a ( 9 a ( 20 a - 49 ) + 406 ) - 735 ) + 175 ) - 21 ) + 1 ) Γ ( a ) 7 w 1 - 7 a ( z - z 0 ) 7 + 1 40320 8 w ( - 5040 a 7 + 13068 a 6 - 13132 a 5 + 6769 a 4 - 1960 a 3 + 322 a 2 - 28 a + 5040 w 7 - 35280 ( a - 1 ) w 6 + 36 ( a - 1 ) ( 2940 a - 2323 ) w 5 - 20 ( a - 1 ) ( 9 a ( 980 a - 1343 ) + 4175 ) w 4 + 2 ( a - 1 ) ( 2 a ( 90 a ( 490 a - 853 ) + 45001 ) - 17729 ) w 3 - 6 ( a - 1 ) ( 2 a - 1 ) ( a ( 60 a ( 147 a - 206 ) + 5891 ) - 947 ) w 2 + ( a - 1 ) ( 2 a - 1 ) ( 3 a - 1 ) ( a ( 10 a ( 588 a - 599 ) + 2097 ) - 247 ) w + 1 ) Γ ( a ) 8 w 1 - 8 a ( z - z 0 ) 8 + /; ( z "\[Rule]" z 0 ) w = Q GammaRegularized - 1 ( a , z 0 ) Set Condition Proportional InverseGammaRegularized a z InverseGammaRegularized a Subscript z 0 -1 w Gamma a w 1 -1 a z -1 Subscript z 0 1 2 2 w -1 a w 1 Gamma a 2 w 1 -1 2 a z -1 Subscript z 0 2 -1 1 6 3 w 2 a 2 -1 4 w 3 a 2 w w 2 1 Gamma a 3 w 1 -1 3 a z -1 Subscript z 0 3 1 24 4 w -6 a 3 18 w 11 a 2 -1 w 18 w 29 6 a w 6 w w 3 11 1 Gamma a 4 w 1 -1 4 a z -1 Subscript z 0 4 -1 1 120 5 w 24 a 4 -1 2 48 w 25 a 3 4 w 36 w 49 35 a 2 -1 2 w w 48 w 121 63 5 a 2 w w 1 12 w w 3 13 1 Gamma a 5 w 1 -1 5 a z -1 Subscript z 0 5 1 720 6 w -120 a 5 600 w 274 a 4 -1 3 400 w 2 474 w 75 a 3 3 w 400 w 2 874 w 399 85 a 2 -1 2 w w w 300 w 1037 923 216 15 a w 2 w w 60 w w 5 437 212 57 1 Gamma a 6 w 1 -1 6 a z -1 Subscript z 0 6 -1 1 5040 7 w 720 w 6 -1 4320 a -1 w 5 36 a -1 300 a -229 w 4 -1 8 a -1 18 a 100 a -129 755 w 3 6 a -1 2 a -1 36 a 25 a -27 269 w 2 -1 24 a a 6 a a 30 a -79 79 -229 54 w 120 w a a a 4 a 9 a 20 a -49 406 -735 175 -21 1 Gamma a 7 w 1 -1 7 a z -1 Subscript z 0 7 1 40320 8 w -5040 a 7 13068 a 6 -1 13132 a 5 6769 a 4 -1 1960 a 3 322 a 2 -1 28 a 5040 w 7 -1 35280 a -1 w 6 36 a -1 2940 a -2323 w 5 -1 20 a -1 9 a 980 a -1343 4175 w 4 2 a -1 2 a 90 a 490 a -853 45001 -17729 w 3 -1 6 a -1 2 a -1 a 60 a 147 a -206 5891 -947 w 2 a -1 2 a -1 3 a -1 a 10 a 588 a -599 2097 -247 w 1 Gamma a 8 w 1 -1 8 a z -1 Subscript z 0 8 Rule z Subscript z 0 w InverseGammaRegularized a Subscript z 0 [/itex]

 Rule Form

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

 2007-05-02