Beta and Airy Functions
06-28-2015, 12:47 AM
Post: #10
 roadrunner Senior Member Posts: 420 Joined: Jun 2015
RE: Beta and Airy Functions
(06-27-2015 12:41 PM)parisse Wrote:  For |z| large, you can use asymptotic series expansion near infinity.
http://people.math.sfu.ca/~cbm/aands/page_448.htm

That's much better. Now it matches this web site: http://keisan.casio.com/exec/system/1180573401 pretty close. I only checked out to +/-100.

Code:
 export c(k) begin  if k == 0 then return 1; end;  return Gamma(3*k+1/2)/54^k/k!/Gamma(k+1/2); end; export d(k) begin  return c(k)*(6*k+1)/(1-6*k);  end; local f(z) begin  local t,t2,k,p,p2;  t:=1;  t2:=0;  p2:=1;  for k from 1 to 80 do   p:=3*k;   p2:=p2*(3*k-2);   t:=t+p2*z^(p)/(p)!;   if t == t2 then return t; end;   t2:=t;  end;  return t; end; local g(z) begin  local t,t2,k,p,p2;  t:=z;  t2:=0;  p2:=1;  for k from 1 to 80 do   p:=3*k+1;   p2:=p2*(3*k-1);   t:=t+p2*z^(p)/(p)!;   if t == t2 then return t; end;   t2:=t;  end;  return t; end; local aismallz(z) begin  return  0.355028053887817*f(z)-  0.258819403792807*g(z); end; local aibigz(z) begin  local s;  if z ≥ 0 then   s:=2/3*z^(3/2);   return π^(−1/2)/2*z^(−1/4)*e^(−s)*   sum((−1)^K*c(K)*s^(−K),K,0,20);  end;  z:=ABS(z);  s:=2/3*z^(3/2);  return π^(−1/2)*z^(−1/4)*(SIN(s+π/4)*   sum((−1)^K*c(2*K)*s^(−2*K),K,0,20)-   COS(s+π/4)*   sum((−1)^K*c(2*K+1)*s^(−2*K-1),K,0,20)); end; local bismallz(z) begin   return (0.355028053887817*f(z)+   0.258819403792807*g(z))*√3; end; local bibigz(z) begin  local s;  if z ≥ 0 then   s:=2/3*z^(3/2);   return π^(−1/2)*z^(−1/4)*e^(s)*    sum(c(K)*s^(−K),K,0,20);  end;  z:=ABS(z);  s:=2/3*z^(3/2);  return π^(−1/2)*z^(−1/4)*(COS(s+π/4)*   sum((−1)^K*c(2*K)*s^(−2*K),K,0,20)+   SIN(s+π/4)*   sum((−1)^K*c(2*K+1)*s^(−2*K-1),K,0,20)); end; export ai(z) begin  HAngle:=0;  if ABS(z) < 10 then   return aismallz(z);  end;  return aibigz(z); end; export bi(z) begin  HAngle:=0;  if ABS(z) < 10 then   return bismallz(z);  end;  return bibigz(z); end;
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 Messages In This Thread Beta and Airy Functions - douganc - 06-24-2015, 11:20 AM RE: Beta and Airy Functions - Tim Wessman - 06-24-2015, 02:39 PM RE: Beta and Airy Functions - Gerald H - 06-24-2015, 03:27 PM RE: Beta and Airy Functions - DrD - 06-24-2015, 03:33 PM RE: Beta and Airy Functions - parisse - 06-24-2015, 06:48 PM RE: Beta and Airy Functions - roadrunner - 06-27-2015, 11:19 AM RE: Beta and Airy Functions - Gerald H - 06-27-2015, 12:15 PM RE: Beta and Airy Functions - parisse - 06-27-2015, 12:41 PM RE: Beta and Airy Functions - roadrunner - 06-28-2015 12:47 AM RE: Beta and Airy Functions - Gerald H - 06-27-2015, 05:25 PM RE: Beta and Airy Functions - douganc - 07-20-2015, 08:10 PM RE: Beta and Airy Functions - roadrunner - 07-20-2015, 08:55 PM

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