How I can define contour integration for FoxâÂÂs H function.
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Let $largeH_m,n^p,q$, the generalized upper incomplete FoxâÂÂs H function
given by
$$
largeH_m,n^p,qleft( z left| beginarraycc (a_1,alpha_1,A_1)cdots (a_p,alpha_p,A_p) \ (a_1,alpha_1,A_1)cdots (b_p,beta_p,B_p) endarray right. right).
$$
$$
frac12pi joint_L frac
prod_i=1^mGamma(b_i+beta_is,B_i)prod_i=1^nGamma(1-a_i-alpha_is,A_i)
prod_i=n+1^pGamma(a_i+alpha_is,A_i) prod_i=m+1^qGamma(1-b_i-beta_is,B_i)
z^-sds.
$$
I use this function and I get
$$
largeH_1,3^2,1left( beta x left| beginarraycc (1,1,0) \ (1,1,beta),(1,1,beta),(0,1,0) endarray right. right).
$$
$$
frac12pi joint_L frac
Gamma(1+s,beta)^2Gamma(-s,0)
Gamma(1-s,0)
(beta x)^-sds.
$$
where $x$ and $beta$ are real positive values ($beta$ diffrent from $beta_j$).
I write the last expression in mathematica.
Now in order to compute this integral the authors said
the contour $L$ is specially chosen such that it is a Mellin-Barnes contour in the complex
s-plane running from $c-iinfty$ to ÿ$c+iinfty$ and the points
$$
s_1=(-b_j-k)/beta_j
$$
for $j=1,2,cdots,m$ and $k=1,2,cdots$ and the points
$$
s_2=(1-a_i+k)/alpha_i
$$
for $i=1,2,cdots,n$ and $k=1,2,cdots$ lie to the left and right of the
chosen contour $L$, respectively.
Now my value are $b_jin1,0$ and $beta_j=1$.
Also $a_i=1$ and $alpha_i=1$.
I run program to see value of $s_1$ and $s_2$, I found that the value of $s_1$ go from $0$ to $-infty$ and value of $s_2$ go from $0$ to $+infty$. Also both of them have one value the same, is $0$.
Now I have problem to define $c$ of $L$.
I would like if it possible how please I can chose this $c$ and $L$.
Thanks.
complex-analysis contour-integration
add a comment |Â
up vote
0
down vote
favorite
Let $largeH_m,n^p,q$, the generalized upper incomplete FoxâÂÂs H function
given by
$$
largeH_m,n^p,qleft( z left| beginarraycc (a_1,alpha_1,A_1)cdots (a_p,alpha_p,A_p) \ (a_1,alpha_1,A_1)cdots (b_p,beta_p,B_p) endarray right. right).
$$
$$
frac12pi joint_L frac
prod_i=1^mGamma(b_i+beta_is,B_i)prod_i=1^nGamma(1-a_i-alpha_is,A_i)
prod_i=n+1^pGamma(a_i+alpha_is,A_i) prod_i=m+1^qGamma(1-b_i-beta_is,B_i)
z^-sds.
$$
I use this function and I get
$$
largeH_1,3^2,1left( beta x left| beginarraycc (1,1,0) \ (1,1,beta),(1,1,beta),(0,1,0) endarray right. right).
$$
$$
frac12pi joint_L frac
Gamma(1+s,beta)^2Gamma(-s,0)
Gamma(1-s,0)
(beta x)^-sds.
$$
where $x$ and $beta$ are real positive values ($beta$ diffrent from $beta_j$).
I write the last expression in mathematica.
Now in order to compute this integral the authors said
the contour $L$ is specially chosen such that it is a Mellin-Barnes contour in the complex
s-plane running from $c-iinfty$ to ÿ$c+iinfty$ and the points
$$
s_1=(-b_j-k)/beta_j
$$
for $j=1,2,cdots,m$ and $k=1,2,cdots$ and the points
$$
s_2=(1-a_i+k)/alpha_i
$$
for $i=1,2,cdots,n$ and $k=1,2,cdots$ lie to the left and right of the
chosen contour $L$, respectively.
Now my value are $b_jin1,0$ and $beta_j=1$.
Also $a_i=1$ and $alpha_i=1$.
I run program to see value of $s_1$ and $s_2$, I found that the value of $s_1$ go from $0$ to $-infty$ and value of $s_2$ go from $0$ to $+infty$. Also both of them have one value the same, is $0$.
Now I have problem to define $c$ of $L$.
I would like if it possible how please I can chose this $c$ and $L$.
Thanks.
complex-analysis contour-integration
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $largeH_m,n^p,q$, the generalized upper incomplete FoxâÂÂs H function
given by
$$
largeH_m,n^p,qleft( z left| beginarraycc (a_1,alpha_1,A_1)cdots (a_p,alpha_p,A_p) \ (a_1,alpha_1,A_1)cdots (b_p,beta_p,B_p) endarray right. right).
$$
$$
frac12pi joint_L frac
prod_i=1^mGamma(b_i+beta_is,B_i)prod_i=1^nGamma(1-a_i-alpha_is,A_i)
prod_i=n+1^pGamma(a_i+alpha_is,A_i) prod_i=m+1^qGamma(1-b_i-beta_is,B_i)
z^-sds.
$$
I use this function and I get
$$
largeH_1,3^2,1left( beta x left| beginarraycc (1,1,0) \ (1,1,beta),(1,1,beta),(0,1,0) endarray right. right).
$$
$$
frac12pi joint_L frac
Gamma(1+s,beta)^2Gamma(-s,0)
Gamma(1-s,0)
(beta x)^-sds.
$$
where $x$ and $beta$ are real positive values ($beta$ diffrent from $beta_j$).
I write the last expression in mathematica.
Now in order to compute this integral the authors said
the contour $L$ is specially chosen such that it is a Mellin-Barnes contour in the complex
s-plane running from $c-iinfty$ to ÿ$c+iinfty$ and the points
$$
s_1=(-b_j-k)/beta_j
$$
for $j=1,2,cdots,m$ and $k=1,2,cdots$ and the points
$$
s_2=(1-a_i+k)/alpha_i
$$
for $i=1,2,cdots,n$ and $k=1,2,cdots$ lie to the left and right of the
chosen contour $L$, respectively.
Now my value are $b_jin1,0$ and $beta_j=1$.
Also $a_i=1$ and $alpha_i=1$.
I run program to see value of $s_1$ and $s_2$, I found that the value of $s_1$ go from $0$ to $-infty$ and value of $s_2$ go from $0$ to $+infty$. Also both of them have one value the same, is $0$.
Now I have problem to define $c$ of $L$.
I would like if it possible how please I can chose this $c$ and $L$.
Thanks.
complex-analysis contour-integration
Let $largeH_m,n^p,q$, the generalized upper incomplete FoxâÂÂs H function
given by
$$
largeH_m,n^p,qleft( z left| beginarraycc (a_1,alpha_1,A_1)cdots (a_p,alpha_p,A_p) \ (a_1,alpha_1,A_1)cdots (b_p,beta_p,B_p) endarray right. right).
$$
$$
frac12pi joint_L frac
prod_i=1^mGamma(b_i+beta_is,B_i)prod_i=1^nGamma(1-a_i-alpha_is,A_i)
prod_i=n+1^pGamma(a_i+alpha_is,A_i) prod_i=m+1^qGamma(1-b_i-beta_is,B_i)
z^-sds.
$$
I use this function and I get
$$
largeH_1,3^2,1left( beta x left| beginarraycc (1,1,0) \ (1,1,beta),(1,1,beta),(0,1,0) endarray right. right).
$$
$$
frac12pi joint_L frac
Gamma(1+s,beta)^2Gamma(-s,0)
Gamma(1-s,0)
(beta x)^-sds.
$$
where $x$ and $beta$ are real positive values ($beta$ diffrent from $beta_j$).
I write the last expression in mathematica.
Now in order to compute this integral the authors said
the contour $L$ is specially chosen such that it is a Mellin-Barnes contour in the complex
s-plane running from $c-iinfty$ to ÿ$c+iinfty$ and the points
$$
s_1=(-b_j-k)/beta_j
$$
for $j=1,2,cdots,m$ and $k=1,2,cdots$ and the points
$$
s_2=(1-a_i+k)/alpha_i
$$
for $i=1,2,cdots,n$ and $k=1,2,cdots$ lie to the left and right of the
chosen contour $L$, respectively.
Now my value are $b_jin1,0$ and $beta_j=1$.
Also $a_i=1$ and $alpha_i=1$.
I run program to see value of $s_1$ and $s_2$, I found that the value of $s_1$ go from $0$ to $-infty$ and value of $s_2$ go from $0$ to $+infty$. Also both of them have one value the same, is $0$.
Now I have problem to define $c$ of $L$.
I would like if it possible how please I can chose this $c$ and $L$.
Thanks.
complex-analysis contour-integration
asked Aug 27 at 16:03
Monir
12
12
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