· Sequences, Series, and Special Functions 627
The graph of the modular surface of r(z), namely, the graph of v =
lr(z)I, is givell'in Fig. 8.17. Some members of the family of curves v =const. (which lie on horizontal planes) and of Arg r( z) = const. are shown
in the figure.
Further properties of the r-function are discussed in Selected Topics,
Sections 4.4 to 4.6.
Theorem 8.49 The B-function has the following properties (with the
restrictions to be indicated below):- B(z, 1) = 1/z, z =F 0
2. B(z,() = B((,z)
- B(z,( + 1) = zhB(z,()
- B(z, () = B(z, ( + n)B((, n)/B(z + (, n), n a positive integer
- B(z,() = B(z + 1,() + B(z,( + 1)
- zB(z, ( + 1) = (B(z + 1, ()
- B(z,() = 2f 0 7r/^2 sin^2 z-^1 ocos^2 C-^1 BdB
- lime_,.ooe'B(x,e)/r(x) = 1, x > 0, e > 1
- B(z, () = r(z)r(()/r(z + ()
Proofs (1) Follows at once from the definition of B(z, ().
(2) By letting t = 1-r in the defining integral (8.20-1) of the B-function,
we have
B(z,()= 1
1(1-r)z-lT(-ldr=B((,z)
provided that Re z > 0 and Re ( > 0. Thus .the B-function is a symmetric
function of z and (.
(3) We haveB(z, ( + 1) = 1
1
tz-^1 (1 - t)C dt = 1
1
tz-^1 (1-t)C-^1 (1-t) dt
= ~(z, () -1
1
tz(l -t)C-l dtprovided that Re ( > -1, Re z > -1. Also, assuming that ( =F 0, z =F O,
integration by parts of the last integral gives1
1. 1 1 11
tz(l-t)C-ldt= -tz(l-t)C I +~ tz-^1 (1-t)Cdt
(^0) z ' 0 ' 0
= CB(z,(+1)
Hence
z
B(z, ( + 1) = B(z, () - CB(z, ( + 1)