About the Project

by parts

AdvancedHelp

(0.005 seconds)

11—20 of 305 matching pages

11: 25.14 Lerch’s Transcendent
25.14.1 Φ ( z , s , a ) n = 0 z n ( a + n ) s , | z | < 1 ; s > 1 , | z | = 1 .
25.14.2 ζ ( s , a ) = Φ ( 1 , s , a ) , s > 1 , a 0 , 1 , 2 , ,
25.14.3 Li s ( z ) = z Φ ( z , s , 1 ) , s > 1 , | z | 1 .
25.14.5 Φ ( z , s , a ) = 1 Γ ( s ) 0 x s 1 e a x 1 z e x d x , s > 1 , a > 0 if z = 1 ; s > 0 , a > 0 if z [ 1 , ) .
25.14.6 Φ ( z , s , a ) = 1 2 a s + 0 z x ( a + x ) s d x 2 0 sin ( x ln z s arctan ( x / a ) ) ( a 2 + x 2 ) s / 2 ( e 2 π x 1 ) d x , a > 0 if | z | < 1 ; s > 1 , a > 0 if | z | = 1 .
12: 26.9 Integer Partitions: Restricted Number and Part Size
§26.9 Integer Partitions: Restricted Number and Part Size
p k ( n ) denotes the number of partitions of n into at most k parts. … … Conjugation establishes a one-to-one correspondence between partitions of n into at most k parts and partitions of n into parts with largest part less than or equal to k . … equivalently, partitions into at most k parts either have exactly k parts, in which case we can subtract one from each part, or they have strictly fewer than k parts. …
13: 5.13 Integrals
5.13.1 1 2 π i c i c + i Γ ( s + a ) Γ ( b s ) z s d s = Γ ( a + b ) z a ( 1 + z ) a + b , ( a + b ) > 0 , a < c < b , | ph z | < π .
5.13.3 1 2 π Γ ( a + i t ) Γ ( b + i t ) Γ ( c i t ) Γ ( d i t ) d t = Γ ( a + c ) Γ ( a + d ) Γ ( b + c ) Γ ( b + d ) Γ ( a + b + c + d ) , a , b , c , d > 0 .
5.13.4 d t Γ ( a + t ) Γ ( b + t ) Γ ( c t ) Γ ( d t ) = Γ ( a + b + c + d 3 ) Γ ( a + c 1 ) Γ ( a + d 1 ) Γ ( b + c 1 ) Γ ( b + d 1 ) , ( a + b + c + d ) > 3 .
5.13.5 1 4 π k = 1 4 Γ ( a k + i t ) Γ ( a k i t ) Γ ( 2 i t ) Γ ( 2 i t ) d t = 1 j < k 4 Γ ( a j + a k ) Γ ( a 1 + a 2 + a 3 + a 4 ) , ( a k ) > 0 , k = 1 , 2 , 3 , 4 .
14: 6 Exponential, Logarithmic, Sine, and
Cosine Integrals
15: 7 Error Functions, Dawson’s and Fresnel Integrals
16: 8 Incomplete Gamma and Related
Functions
17: 12 Parabolic Cylinder Functions
18: 28 Mathieu Functions and Hill’s Equation
19: 30 Spheroidal Wave Functions
20: 33 Coulomb Functions