Page 154 - Wind Energy Handbook

128 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

ð
1
P n (v)P k dv ¼ 0 n 6¼ k (3:152)
1
For n ¼ 0 the Legendre polynomial of the second kind is

1 1 þ v
Q 0 (v) ¼ ln (3:153)
2 1 v
For n . 0 the Legendre polynomials of the second kind Q n (v) can be obtained from
the polynomials of the first kind. For n ¼ 1to4

Q 1 (v) ¼ (P 1 (v)Q 0 (v)Þ 1

3
Q 2 (v) ¼ (P 2 (v)Q 0 (v)Þ v (3:154)
2
5 2 2
Q 3 (v) ¼ (P 3 (v)Q 0 (v)Þ v þ
2 3
35 3 55
Q 4 (v) ¼ (P 4 (v)Q 0 (v)Þ v þ v
8 24

The solutions for Equation (3.149b), with m ¼ 0, are the same as for (3.149a) but
with imaginary arguments, i.e.,

Ö 2 (ç) ¼ P n (iç) (3:155a)

and

Ö 2 (ç) ¼ Q n (iç) (3:155b)

where

1
Q 0 (iç) ¼ i tan (ç) for ç , 1

and

ð
Q 0 (iç) ¼ i tan 1 ç for ç . 1
2

For non-zero values of the integer m the solutions of Equation (3.149a) become

d m
m
2 m=2
P (v) ¼ (1 v ) P n (v) (3:156)
n m
dv
m
If m . n then P (v) ¼ 0.
n