Why is this function a really good asymptotic for exp(x)sqrt(x) f(x)=sum_(n=0)^(oo) a_n x^n \ a_n = (1)/(Gamma(n+0.5)) exp(x)sqrt(x), where for large positive numbers, f(x)exp(−x)~~sqrt x?

firmezas1 2022-10-03 Answered
Why is this function a really good asymptotic for exp ( x ) x
f ( x ) = n = 0 a n x n a n = 1 Γ ( n + 0.5 )
Why is this entire function a really good asymptotic for exp ( x ) x , where for large positive numbers, f ( x ) exp ( x ) x ?
As |x| gets larger, the error term is asymptotically f ( x ) exp ( x ) x 1 x Γ ( 0.5 ) , and the error term for f ( x ) exp ( x ) x exp ( x ) x Γ ( 0.5 ) . If we treat f ( x ) as an infinite Laurent series, than it does not converge.
I stumbled upon the result, using numerical approximations, so I can't really explain the equation for the a n coefficients, other than it appears to be the numerical limit of a pseudo Cauchy integral for the a n coefficients as the circle for the Cauchy integral path gets larger. I suspect the formula has been seen before, and can be generated by some other technique. By definition, for any entire function f ( x ), we have for any value of real r:
a n = x n f ( x ) = π π 1 2 π ( r e i x ) n f ( r e i x ) ) d x
The conjecture is that this is an equivalent definition for a n , where f ( x ) exp ( x ) x and x r e i x
a n = lim r π π 1 2 π ( r e i x ) n exp ( r e i x ) r e i x ) d x = 1 Γ ( n + 0.5 )
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

oldgaffer1b
Answered 2022-10-04 Author has 9 answers
Repeated integrations by parts show that, for every positive a and x
0 x e t t a 1 d t = Γ ( a ) e x n 0 x n + a Γ ( n + a + 1 ) .
When x , the LHS converges to Γ ( a ), hence the series in the RHS is equivalent to e x . Now,
n 0 x n Γ ( n + a ) = 1 Γ ( a ) + x 1 a n 0 x n + a Γ ( n + a + 1 )
hence
n 0 x n Γ ( n + a ) x 1 a e x .
For a = 1 2 , this is the result mentioned in the question.
An exact formula using the incomplete gamma function γ ( a ,   ) (that is, the LHS of the first identity in this answer) is
n 0 x n Γ ( n + a ) = γ ( a , x ) Γ ( a ) x 1 a e x + 1 Γ ( a ) .
Edit: ...And this approach yields the more precise expansion, also mentioned in the question,
n 0 x n Γ ( n + a ) = x 1 a e x + 1 a Γ ( a ) 1 x + O ( 1 x 2 ) .
More generally, for every nonnegative integer N and every noninteger a.
n 0 x n Γ ( n + a ) = x 1 a e x + sin ( π a ) π k = 1 N Γ ( k + 1 a ) x k + O ( 1 x N + 1 ) .
Did you like this example?
Subscribe for all access
KesseTher12
Answered 2022-10-05 Author has 6 answers
In fact the story starts with finding an asymptotic expansion for the function
f ( x ) = n = 0 x n Γ ( n + 1 / 2 ) = π x e x e r f ( x ) + 1 π
which is given by
f ( x ) x e x
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-08-17
Evaluate the maximum rate of change of f(x,y)=ln(x2+y2) at the point (3,2) and the direction in which it occurs.
What is maximum rate of change and direction(unit vector) in which it occurs?
asked 2021-12-12
Solve, please, (12)x=162.
asked 2022-07-07
Prove that log 2 5 + log 2 7 > log 12
What I tried so far:
log 2 5 + log 2 7 > log 3 + log 4
( log 5 + log 7 ) 2 2 log 5 log 7 > log 3 + log 4
But it seems that I'm not even near the result.
Every suggestion / hint would be appreciated :)
Thanks in advance.
EDIT: log means log 10
asked 2022-11-24
Finding the interval of which a multivariable function is defined
Find the interval in which f ( x , y , z ) = z + l n ( 1 x 2 y 2 ) is defined
So all that is need to to check for which values l n ( 1 x 2 y 2 ) 0
That mean 1 x 2 y 2 1 x 2 + y 2 0
But how do I find the specific x terms and y terms?
Or is x 2 + y 2 0 is sufficient?
asked 2022-03-22
Solve the exponential equation
52x+1+5x4=0.
asked 2022-09-06
If log 2 3 = a and log 5 2 = b then log 24 50 is equal to?
I guess this has to be done by using simple logarithmic rules, but I do not how to start. Answer in my booklet is b + 2 b ( a + 3 )
asked 2022-07-10
How do I solve for x in ln ( x ) ln ( x ) = 2 + ln ( x )