do we have relation between LommelS2 and MeijerG in matlab?

Do we have matlab relation between two special function "LommelS2 and MeijerG"?

How to convert LommelS2 code into MeijerG?

Please help !!

Answer 1

[Walter Roberson]

https://www.maplesoft.com/support/help/Maple/view.aspx?path=LommelS1

and scroll down to near the bottom to see the conversions.

[gourav pandey]

thank you for you reply

n=2;
m=5;
omega=10;
a=-(1./4).*n-(1./4).*m+5./4
k2=(1./4).*n+(1./4).*m+1./4
k3=-(1./4).*n-(1./4).*m-1./4
k4=-(1./4).*n-(1./4).*m+5./4
z=-(1./4).*omega.^2
fun=meijerG([[a], []], [[k2,k3,k4], []], z);
LommelS2 = -fun./(2.*sqrt(pi).*gamma((1./2).*n+(1./2).*m).*2.^((1./2).*n+(1./2).*m-1./2));

## can you tell me what is wrong in this code??

[Walter Roberson]

format long g
n=2;
m=5;
omega=10;
a=-(1./4).*n-(1./4).*m+5./4

a = 
                      -0.5

k2=(1./4).*n+(1./4).*m+1./4

k2 = 
     2

k3=-(1./4).*n-(1./4).*m-1./4

k3 = 
    -2

k4=-(1./4).*n-(1./4).*m+5./4

k4 = 
                      -0.5

z=-(1./4).*omega.^2

z = 
   -25

fun=meijerG([a], [], [k2,k3,k4], [], z);
LommelS2 = -fun./(2.*sqrt(pi).*gamma((1./2).*n+(1./2).*m).*2.^((1./2).*n+(1./2).*m-1./2));

LommelS2

LommelS2 = 
       2.52409581072613e-06 +  0.000920671745445133i

[gourav pandey]

Thank you Walter.

[Walter Roberson]

Is the question resolved?

[gourav pandey]

actually not completely.

fun1:=evalf(convert(LommelS2(-(1/2)*n-(1/2)*j+3/2, (1/2)*n+(1/2)*j+1/2, I*omega),MeijerG))assuming omega::real,0<omega;

fun1 := 0.00004212956333 + 0.00001923699304 I

fun2:=evalf(convert(LommelS2(-(1/2)*n-(1/2)*j+3/2, (1/2)*n+(1/2)*j+1/2, -I*omega),MeijerG))assuming omega::real,0<omega;

fun2 := 0.00004212956333 + 0.00001923699304 I

fun3:=evalf(LommelS2(-(1/2)*n-(1/2)*j+3/2, (1/2)*n+(1/2)*j+1/2, -I*omega));

fun3 := 0.00004212956330 - 0.00001923699311 I

fun4:=evalf(LommelS2(-(1/2)*n-(1/2)*j+3/2, (1/2)*n+(1/2)*j+1/2, I*omega));

fun4 := 0.4212956330e-4+0.1923699311e-4*I

as you suggested to get relation from MAPLE. so the above code is coded in maple. fun1 and fun2 are conjugate of eact other similarly fun3 and fun4. but we can see the difference in the answer of fun2 and fun3. you can see in BOLD letters.

fun1 and fun2 are converting the lommelS2 into MeijerG where fun3 and fun4 are directing calculating the lommelS2.

there is an issue with conjugate no. in MeijerG. do you know how to resolve this issue??

thank you!!

[Walter Roberson]

Your output of a specific numeric number for fun1 implies that you were using specific numeric n and specific numeric omega (even though you have an assumption as-if omega were a variable instead of constant.) . However, I was not able to figure out what specific numeric n and omega you were using.

Is it correct that your j in the Maple code refers to sqrt(-1) ?

[Walter Roberson]

Maple does indeed produce inconsistent results for the conversion depending upon whether the z is positive imaginary or negative imaginary. I have filed a bug report for Maple and will see what the response is.

[gourav pandey]

"n" and "j" are the constant where omega is variable between -infinity to +infinty. actually bymistakenly i forget the write the value of n,j,omega. but here my point is "why we are getting different answer for same values of n,j and omega"

even you can check the above code for any random values of n,j,omega.

'j' is not an sqrt(-1)

[Mubashara Wali]

I need the Matlab code for LommelS2 function. Can you help me?

[Walter Roberson]

syms mu nu z
numer =  meijerG([mu/2 + 1/2], [], [mu/2 + 1/2, nu/2, -nu/2], [], z.^2/4) .* 2.^mu;
denom = 2 .* gamma(-mu/2 + nu/2 + 1/2) .* gamma(-mu/2 - nu/2 + 1/2);
S2 = numer./denom;
LommelS2(nu, nu, z) = S2;

Caution: this might not operate properly if mu or nu are not scalars. Also, there could be problems if nu = mu+1

[Mubashara Wali]

Thanx Walter Roberson

[Mubashara Wali]

Walter Roberson its not working, giving a error "Attempt to execute SCRIPT Lommel as a function:"

Actually I need to find the value of the following

where 0 <alpha <1

ABC(alpha)=1- alpha + (alpha / gamma(alpha))

and t belongs to [0, 1]

Your kind help will be appreciated.

[Walter Roberson]

Could you post a link to where I suggested creating or calling a function or script named Lommel ? The code I posted at https://www.mathworks.com/matlabcentral/answers/726363-do-we-have-relation-between-lommels2-and-meijerg-in-matlab#comment_1437812 shows creating a symbolic function named LommelS2

Your "actually I need" uses LommelS1 not LommelS2 so I am confused.

[Walter Roberson]

syms mu nu z

S1part1 = meijerG([mu/2 + 1/2], [], [mu/2 + 1/2], [nu/2, -nu/2], z^2/4);
S1part2 = 2.^(mu - 1);
S1part3 = 2 .* gamma(mu/2 + 1/2 + nu/2) .* gamma(mu/2 + 1/2 - nu/2);
LommelS1(mu, nu, z) = simplify(S1part1 .* S1part2 .* S1part3)

LommelS1(mu, nu, z) = 

S2part1 =  meijerG([mu/2 + 1/2], [], [mu/2 + 1/2, nu/2, -nu/2], [], z.^2/4);
S2part2 = 2.^mu;
S2part3 = 2 .* gamma(-mu/2 + nu/2 + 1/2) .* gamma(-mu/2 - nu/2 + 1/2);
LommelS2(mu, nu, z) = simplify(S2part1 .* S2part2 ./ S2part3)

LommelS2(mu, nu, z) = 

syms alpha t real
assume(0 < alpha & alpha < 1)
assume(0 <= t & t <= 1)
ABC(alpha) = 1 - alpha + (alpha / gamma(alpha))  %CORRECTED

ABC(alpha) = 

ypart1 = (1-alpha)/ABC(alpha);
ypart2 = gamma(alpha+1) * ABC(alpha);
ypart3 = t.^(1+alpha) - sqrt(t) * LommelS1(3/2+alpha, 1/2, t)

ypart3 = 

y(t) = simplify(ypart1 + ypart3 ./ ypart2)

y(t) = 

[Mubashara Wali]

Walter Roberson thanx for detailed answer. It made my concept clear now

[Walter Roberson]

it looks like I entered the wrong ABC; I will fix it when I get back to my computer.

[Mubashara Wali]

Ok

[Walter Roberson]

I have corrected the ABC and regenerated the result. I had accidentally put in 1 - 1- alpha instead of 1 - alpha

[Mubashara Wali]

Thankyou

Answer 2

[gourav pandey]

Walter Roberson, thanks for your reply..can you help me here also

clc;
close;
clear all;
n=5;
h=@(omega1) 0;
   for m= 0:1:n
        mu1c = -(1./2).*n-(1./2).*m+3./2 ;
        nu1c =  (1./2).*n+(1./2).*m+1./2 ; 
       Lommels =@(omega1) meijerG([(1./2).*mu1c+1./2], [], [(1./2).*mu1c+1./2, (1./2).*nu1c, -(1./2).*nu1c], [], -(1./4).*omega1.^2).*2^mu1c./(2.*gamma(-(1./2).*mu1c+(1./2).*nu1c+1./2).*gamma(-(1./2).*mu1c-(1./2).*nu1c+1./2))
   end
Lommels(1)
Lommels(10)
Lommels(99)
Lommels(100)
Lommels(500)
Lommels(1000)

%% Why m i getting sudden blow in value of 'Lommels' when omega1 veluse more than 120 or 150... it should decay as omega1 increase.

Thank you in advance!!

[Walter Roberson]

You are overwriting Lommels within the loop, so the final result will reflect only m == n