PROPERTIES OF SINE WAVE PLUS NOISE 115 



where / = y^/(a^ — /) and P2n( ) denotes the Legendre polynomial of 

 order 2n. We have written this as an asymptotic expansion because it 

 obviously is one when y, and hence /, is zero in which case 



n 1.3.5 ••.(2» - 1) 



P2n(0) = (-)' 



2.4.... 2» 



When y is near a or is greater than o, a suitable asymptotic expansion may 

 be obtained from the second integral in (1.13) by expanding {2a — z)~^'^ 

 in powers of z/(2a) and integrating termwise. The upper limit of integra- 

 tion, 2a, may be replaced by oc since <p{z -\- y — a) may be assumed to be 

 negligibly small when z exceeds 2a. We thus obtain 

 1 * ('l^ / 1 \"+i/2 /•« 



P.(y) ~ - E ^ (;i ) fi^ + y- a)z'"'"' dz 



T n=o n I \Za/ Jo 



/ \ <» /1\ / 1 \n+l/2 /.» \ '^ ) 



^ (p{y — a) y^ {^Jin ( JL ) ■ / g-^(v-a)-(^2/2) ^n-m ^^ 

 IT n=o nl \2a/ Jo 



where we have used the notation 



(«)o = 1, {a)n = a{a+l) • • • {a + n - 1). 



The integrals occurring in (1.15) are related to the parabolic cylinder 

 function^ Dm(x). Their properties may be obtained from the known 

 properties of these functions or may be obtained by working directly with 

 the integrals. 



Suppose now that a is very large so that only the leading term in the series 

 (1.15) for pi(y) need be retained. 



Then 



piiy) ^ a-^'^ F(y - a) (1.16) 



where 



F(s) = IT-' 2-'" f <p(z + s)z-"' dz (1.17) 



Jo 



By writing out (p(z + s), expanding exp (—zs) in a power series, and inte- 

 grating termwise we see that 



Fis) = ^?«1^^ t ^^j^ i-sViy ■ (1.18) 



= i2T)-'s"\(s/V~2)K^{sV^) 

 where K denotes a modified Bessel function.^ The relation (1.18) may also 



6 Whittaker and Watson, "Modern Analysis," 4th ed. (1927), 347-351. 



7 A table of A^Ct) is given by H. Carsten and N. McKerrow, Phil. Mag. S7, Vol. 35 

 (1944), 812-818. 



