inverse LaPlace transform of confluent hypergeometric function

hello, I want to calculate the inverse laplace transform of a characteristic function

this is the fourier transform of h(x), which is converted to Laplace transform with p = -i*t

from here we make a characteristic function which is

this characteristic function is converted to hypergeometric function

I have to get the inverse laplace transform of this function, I have written the following code:

N = 3; x_bar = 2.5; a = 1.3; b = 13; c = a+b;
syms s t
p = -1*1j*s

p = 

z = -1*((N*x_bar*(c/a)*p)/(p+1))

z = 

h = hypergeom(a,c,z)

h = 

C_slow = h/((p+1)^N)

C_slow = 

f(t) = ilaplace(C_slow)

f(t) = 

output of code:

z =

-(s*165i)/(2*(- 1 + s*1i))

h =

hypergeom(13/10, 143/10, -(s*165i)/(2*(- 1 + s*1i)))

do we have to put the vlaue of ''s'' to evalute the hypergeom function? I don't know what i am missing here...any help would be appreciated

[Paul]

Hi Muhammad,

Doesn't the paper cited below state that the inverse Laplace of the C_N(p)_slow is unknown and numerical methods are required to solve the problem as discussed in Section 4 of that paper?

[Muhammad Abdullah]

Yes, the paper mentioned as you stated, (they mentioned how they solved the confluent function and ilaplace via numerical methods)

but now we have a command available for hypergeometric function, which is going to give us a function in terms of s (only if my z part in Cn(p)_slow is correct.) which is the boxed part in the picture, after that I can assume that whole Cn(p)_slow is correct. So I am not sure whether my code is correctly making up the whole Cn(p)_slow function or not...

and we still have ilaplace in matlab, which is surely able to solve the inverse laplace problem also... I am not sure about making of the Cn(p)_slow...

Answer 1

[Umar]

Brother Muhammad,

You asked, do we have to put the vlaue of ''s'' to evalute the hypergeom function? I don't know what i am missing here...any help would be appreciated

The answer is yes, you need to assign a specific value to the variable s before evaluating the hypergeometric function h because the hypergeometric function requires numerical values for its parameters to evaluate correctly.

>> % Corrected Code N = 3; x_bar = 2.5; a = 1.3; b = 13; c = a + b; syms s t p = -1*1j*s; z = -1*((N*x_bar*(c/a)*p)/(p+1));

% Assign a specific value to s s_val = 2; % Example value for s z_val = subs(z, s, s_val); h = hypergeom(a, c, z_val); C_slow = h/((p+1)^N); f(t) = ilaplace(C_slow);

% Display test results s_val z_val h f(t)

Hope, this will help resolve your issue.

[Muhammad Abdullah]

Thank you very much Umar... Actually i need to calculate the probability of detection (Pd) from the inverse laplace transform. in the article, it is mentioned that hypergeometric function makes a series... but arg(z) should be 0<= arg(z) <pi/2, so I don't know what could be the value of s for that. in continuation to the previously mentioned characteristic function, the Pd is calculated by taking its inverse laplace transform and then integrating this function over an interval.

this will give you some real values of Pd for certain signal SNR values. they use the beta distribution for calculation of Pd. any help from RCS community(radar cross section) guy would be appreciated...regards