• 5

NUMERICAL AND ALGEBRAIC

Gain in decibels of P2 relative to Pi

G = 101og10(P2/Pi).

To within two percent

(2tt)1/2 « 2.5; 7T2 « 10; e3 « 20; 210 « 10'

Euler-Mascheroni constant1 7 = 0.57722

Gamma Function F(x + 1) = xT(x):

r(i/6)

= 5.5663

r(3/5)

= 1.4892

r(i/5)

= 4.5908

T(2/3)

= 1.3541

r(i/4)

= 3.6256

T(3/4)

= 1.2254

r(i/3)

= 2.6789

T(4/5)

= 1.1642

r(2/5)

= 2.2182

r(5/6)

= 1.1288

r(i/2)

= 1.7725 = V^r

r(i)

= 1.0

Binomial Theorem (good for | x |< 1 or a = positive integer):

00

/1 ,                   V^ fa\ k _ , ,                         , a(a ~ !) 2 .                        ~ 1)(Q! ~ 2) 3 .

(1 + x) = >                       x = 1 + ax H--------------------- x H------------------------------ x + . . .

v ) Z^ Vfc/                                                               2!                                 3!

fc = 0

Rothe-Hagen identity2 (good for all complex x, y, z except when singular):

n

x /x + kz\                                                         y              /y + (n —

Ex /x + kz\                                   y              /y(n — k)z\

x + kz ^ k ' y + (n — k)z \ n — k '

k=0

x + /x + y + nz

/x ^ y ^ nz\ V n, /

x + 2/ + nz ^ n

Newberger's summation formula3 [good for jit, nonintegral, Re (a + /3) > — 1]:

00

E(-l)n Ja_7n(z) J/3 + 7n(z)                                       7T

n n                                                                                sm fin

n = — 00

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