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11: 7.10 Derivatives
7.10.1 d n + 1 erf z d z n + 1 = ( - 1 ) n 2 π H n ( z ) e - z 2 , n = 0 , 1 , 2 , .
7.10.2 w ( z ) = - 2 z w ( z ) + ( 2 i / π ) ,
12: 6.4 Analytic Continuation
6.4.2 E 1 ( z e 2 m π i ) = E 1 ( z ) - 2 m π i , m ,
6.4.4 Ci ( z e ± π i ) = ± π i + Ci ( z ) ,
6.4.5 Chi ( z e ± π i ) = ± π i + Chi ( z ) ,
6.4.6 f ( z e ± π i ) = π e i z - f ( z ) ,
6.4.7 g ( z e ± π i ) = π i e i z + g ( z ) .
13: 24.11 Asymptotic Approximations
24.11.1 ( - 1 ) n + 1 B 2 n 2 ( 2 n ) ! ( 2 π ) 2 n ,
24.11.2 ( - 1 ) n + 1 B 2 n 4 π n ( n π e ) 2 n ,
24.11.3 ( - 1 ) n E 2 n 2 2 n + 2 ( 2 n ) ! π 2 n + 1 ,
24.11.4 ( - 1 ) n E 2 n 8 n π ( 4 n π e ) 2 n .
24.11.5 ( - 1 ) n / 2 - 1 ( 2 π ) n 2 ( n ! ) B n ( x ) { cos ( 2 π x ) , n  even , sin ( 2 π x ) , n  odd ,
14: 10.64 Integral Representations
10.64.1 ber n ( x 2 ) = ( - 1 ) n π 0 π cos ( x sin t - n t ) cosh ( x sin t ) d t ,
10.64.2 bei n ( x 2 ) = ( - 1 ) n π 0 π sin ( x sin t - n t ) sinh ( x sin t ) d t .
15: 4.4 Special Values and Limits
4.4.2 ln ( - 1 ± i 0 ) = ± π i ,
4.4.3 ln ( ± i ) = ± 1 2 π i .
4.4.5 e ± π i = - 1 ,
4.4.6 e ± π i / 2 = ± i ,
16: 24.7 Integral Representations
24.7.2 B 2 n = ( - 1 ) n + 1 4 n 0 t 2 n - 1 e 2 π t - 1 d t = ( - 1 ) n + 1 2 n 0 t 2 n - 1 e - π t csch ( π t ) d t ,
24.7.3 B 2 n = ( - 1 ) n + 1 π 1 - 2 1 - 2 n 0 t 2 n sech 2 ( π t ) d t ,
24.7.4 B 2 n = ( - 1 ) n + 1 π 0 t 2 n csch 2 ( π t ) d t ,
24.7.5 B 2 n = ( - 1 ) n 2 n ( 2 n - 1 ) π 0 t 2 n - 2 ln ( 1 - e - 2 π t ) d t .
24.7.6 E 2 n = ( - 1 ) n 2 2 n + 1 0 t 2 n sech ( π t ) d t .
17: 6.9 Continued Fraction
6.9.1 E 1 ( z ) = e - z z + 1 1 + 1 z + 2 1 + 2 z + 3 1 + 3 z + , | ph z | < π .
18: 7.7 Integral Representations
7.7.1 erfc z = 2 π e - z 2 0 e - z 2 t 2 t 2 + 1 d t , | ph z | 1 4 π ,
7.7.2 w ( z ) = 1 π i - e - t 2 d t t - z = 2 z π i 0 e - t 2 d t t 2 - z 2 , z > 0 .
7.7.3 0 e - a t 2 + 2 i z t d t = 1 2 π a e - z 2 / a + i a F ( z a ) , a > 0 .
7.7.4 0 e - a t t + z 2 d t = π a e a z 2 erfc ( a z ) , a > 0 , z > 0 .
7.7.9 0 x erf t d t = x erf x + 1 π ( e - x 2 - 1 ) .
19: 4.19 Maclaurin Series and Laurent Series
4.19.3 tan z = z + z 3 3 + 2 15 z 5 + 17 315 z 7 + + ( - 1 ) n - 1 2 2 n ( 2 2 n - 1 ) B 2 n ( 2 n ) ! z 2 n - 1 + , | z | < 1 2 π ,
4.19.4 csc z = 1 z + z 6 + 7 360 z 3 + 31 15120 z 5 + + ( - 1 ) n - 1 2 ( 2 2 n - 1 - 1 ) B 2 n ( 2 n ) ! z 2 n - 1 + , 0 < | z | < π ,
4.19.5 sec z = 1 + z 2 2 + 5 24 z 4 + 61 720 z 6 + + ( - 1 ) n E 2 n ( 2 n ) ! z 2 n + , | z | < 1 2 π ,
4.19.6 cot z = 1 z - z 3 - z 3 45 - 2 945 z 5 - - ( - 1 ) n - 1 2 2 n B 2 n ( 2 n ) ! z 2 n - 1 - , 0 < | z | < π ,
4.19.7 ln ( sin z z ) = n = 1 ( - 1 ) n 2 2 n - 1 B 2 n n ( 2 n ) ! z 2 n , | z | < π ,
20: 24.9 Inequalities
24.9.4 2 ( 2 n + 1 ) ! ( 2 π ) 2 n + 1 > ( - 1 ) n + 1 B 2 n + 1 ( x ) > 0 , n = 2 , 3 , ,
24.9.5 4 ( 2 n - 1 ) ! π 2 n 2 2 n - 1 2 2 n - 2 > ( - 1 ) n E 2 n - 1 ( x ) > 0 .
24.9.6 5 π n ( n π e ) 2 n > ( - 1 ) n + 1 B 2 n > 4 π n ( n π e ) 2 n ,
24.9.9 β = 2 + ln ( 1 - 6 π - 2 ) ln 2 = 0.6491 .
24.9.10 4 n + 1 ( 2 n ) ! π 2 n + 1 > ( - 1 ) n E 2 n > 4 n + 1 ( 2 n ) ! π 2 n + 1 1 1 + 3 - 1 - 2 n .