Inverse Laplace transform related to modified Bessel function of the second kind
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To solve a fluid diffusion problem, I need to calculate an inverse Laplace transform. The integral, according to the Inverse Laplace theorem, has the form:
beginequation labeli_laplace_1
p(r,t) = fracp_c2pi ilim_n rightarrow inftyint_1-ibeta_n^1+ibeta_nfrace^stsfracK_0(sqrtfracskappa r)K_0(sqrtfracskappa a) d s
endequation
where $p_c,r,a$ can be seen as constants. I choose a contour as in attached Figure 1 to avoid the branch point $z=0$ during the computation. The integral is then instead calculated on the contour including $C_R,C_rho,Gamma_1,Gamma_2$. On $C_rho$, in several textbooks and papers I saw an approximation as followed:
beginequation labeli_laplace_form
lim_rho rightarrow 0int_C_rho frace^stsfracK_0(sqrtfracskappa r)K_0(sqrtfracskappa a) ds
approx lim_rho rightarrow 0int_C_rho frace^sts d s
= 2pi i
endequation
However, I still do not fully understand how good is such an approximation and whether it is mathematically rigorous. Can you give me some insightful discussion?
bessel-functions inverselaplace
asked Aug 8 at 17:05
Hoang Nguyen
11
add a comment |Â
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0
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To solve a fluid diffusion problem, I need to calculate an inverse Laplace transform. The integral, according to the Inverse Laplace theorem, has the form:
beginequation labeli_laplace_1
p(r,t) = fracp_c2pi ilim_n rightarrow inftyint_1-ibeta_n^1+ibeta_nfrace^stsfracK_0(sqrtfracskappa r)K_0(sqrtfracskappa a) d s
endequation
where $p_c,r,a$ can be seen as constants. I choose a contour as in attached Figure 1 to avoid the branch point $z=0$ during the computation. The integral is then instead calculated on the contour including $C_R,C_rho,Gamma_1,Gamma_2$. On $C_rho$, in several textbooks and papers I saw an approximation as followed:
beginequation labeli_laplace_form
lim_rho rightarrow 0int_C_rho frace^stsfracK_0(sqrtfracskappa r)K_0(sqrtfracskappa a) ds
approx lim_rho rightarrow 0int_C_rho frace^sts d s
= 2pi i
endequation
However, I still do not fully understand how good is such an approximation and whether it is mathematically rigorous. Can you give me some insightful discussion?
bessel-functions inverselaplace
asked Aug 8 at 17:05
Hoang Nguyen
11
Have you tried writing out the bessel functions in their Laurant series around $s=0$?
– Alex R.
Aug 8 at 17:14
Hi Alex, I did. As a real-valued function, $K_0(x)$ converges to $-ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in the principal branch cut, z = 0 is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 17:36
1
Have a look at this: math.stackexchange.com/questions/480235/…
– Ron Gordon
Aug 8 at 17:52
Thanks for the answer, Ron. I have a question related to the integral on $C_4$. Can you tell me more details how did you compute this, and is this an exact or approximate solution? My concern in this step is, as a real-valued function, $K_0(x)$ converges to $−ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in a branch cut, $z = 0$ is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 19:03
@HoangNguyen: on $C_4$, there is a ratio of two logs of very small numbers which approaches one. (Recall that we are integrating a ratio of Bessels.) The branch cut is defined by the given contour.
– Ron Gordon
Aug 8 at 19:09
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
To solve a fluid diffusion problem, I need to calculate an inverse Laplace transform. The integral, according to the Inverse Laplace theorem, has the form:
beginequation labeli_laplace_1
p(r,t) = fracp_c2pi ilim_n rightarrow inftyint_1-ibeta_n^1+ibeta_nfrace^stsfracK_0(sqrtfracskappa r)K_0(sqrtfracskappa a) d s
endequation
where $p_c,r,a$ can be seen as constants. I choose a contour as in attached Figure 1 to avoid the branch point $z=0$ during the computation. The integral is then instead calculated on the contour including $C_R,C_rho,Gamma_1,Gamma_2$. On $C_rho$, in several textbooks and papers I saw an approximation as followed:
beginequation labeli_laplace_form
lim_rho rightarrow 0int_C_rho frace^stsfracK_0(sqrtfracskappa r)K_0(sqrtfracskappa a) ds
approx lim_rho rightarrow 0int_C_rho frace^sts d s
= 2pi i
endequation
However, I still do not fully understand how good is such an approximation and whether it is mathematically rigorous. Can you give me some insightful discussion?
bessel-functions inverselaplace
asked Aug 8 at 17:05
Hoang Nguyen
11
To solve a fluid diffusion problem, I need to calculate an inverse Laplace transform. The integral, according to the Inverse Laplace theorem, has the form:
beginequation labeli_laplace_1
p(r,t) = fracp_c2pi ilim_n rightarrow inftyint_1-ibeta_n^1+ibeta_nfrace^stsfracK_0(sqrtfracskappa r)K_0(sqrtfracskappa a) d s
endequation
where $p_c,r,a$ can be seen as constants. I choose a contour as in attached Figure 1 to avoid the branch point $z=0$ during the computation. The integral is then instead calculated on the contour including $C_R,C_rho,Gamma_1,Gamma_2$. On $C_rho$, in several textbooks and papers I saw an approximation as followed:
beginequation labeli_laplace_form
lim_rho rightarrow 0int_C_rho frace^stsfracK_0(sqrtfracskappa r)K_0(sqrtfracskappa a) ds
approx lim_rho rightarrow 0int_C_rho frace^sts d s
= 2pi i
endequation
However, I still do not fully understand how good is such an approximation and whether it is mathematically rigorous. Can you give me some insightful discussion?
bessel-functions inverselaplace
asked Aug 8 at 17:05
Hoang Nguyen
11
edited Aug 8 at 17:38
asked Aug 8 at 17:05
Hoang Nguyen
11
asked Aug 8 at 17:05
Hoang Nguyen
11
asked Aug 8 at 17:05
Hoang Nguyen
11
11
Have you tried writing out the bessel functions in their Laurant series around $s=0$?
– Alex R.
Aug 8 at 17:14
Hi Alex, I did. As a real-valued function, $K_0(x)$ converges to $-ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in the principal branch cut, z = 0 is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 17:36
1
Have a look at this: math.stackexchange.com/questions/480235/…
– Ron Gordon
Aug 8 at 17:52
Thanks for the answer, Ron. I have a question related to the integral on $C_4$. Can you tell me more details how did you compute this, and is this an exact or approximate solution? My concern in this step is, as a real-valued function, $K_0(x)$ converges to $−ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in a branch cut, $z = 0$ is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 19:03
@HoangNguyen: on $C_4$, there is a ratio of two logs of very small numbers which approaches one. (Recall that we are integrating a ratio of Bessels.) The branch cut is defined by the given contour.
– Ron Gordon
Aug 8 at 19:09
add a comment |Â
Have you tried writing out the bessel functions in their Laurant series around $s=0$?
– Alex R.
Aug 8 at 17:14
Hi Alex, I did. As a real-valued function, $K_0(x)$ converges to $-ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in the principal branch cut, z = 0 is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 17:36
1
Have a look at this: math.stackexchange.com/questions/480235/…
– Ron Gordon
Aug 8 at 17:52
Thanks for the answer, Ron. I have a question related to the integral on $C_4$. Can you tell me more details how did you compute this, and is this an exact or approximate solution? My concern in this step is, as a real-valued function, $K_0(x)$ converges to $−ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in a branch cut, $z = 0$ is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 19:03
@HoangNguyen: on $C_4$, there is a ratio of two logs of very small numbers which approaches one. (Recall that we are integrating a ratio of Bessels.) The branch cut is defined by the given contour.
– Ron Gordon
Aug 8 at 19:09
Have you tried writing out the bessel functions in their Laurant series around $s=0$?
– Alex R.
Aug 8 at 17:14
Have you tried writing out the bessel functions in their Laurant series around $s=0$?
– Alex R.
Aug 8 at 17:14
Hi Alex, I did. As a real-valued function, $K_0(x)$ converges to $-ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in the principal branch cut, z = 0 is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 17:36
Hi Alex, I did. As a real-valued function, $K_0(x)$ converges to $-ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in the principal branch cut, z = 0 is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 17:36
1
1
Have a look at this: math.stackexchange.com/questions/480235/…
– Ron Gordon
Aug 8 at 17:52
Have a look at this: math.stackexchange.com/questions/480235/…
– Ron Gordon
Aug 8 at 17:52
Thanks for the answer, Ron. I have a question related to the integral on $C_4$. Can you tell me more details how did you compute this, and is this an exact or approximate solution? My concern in this step is, as a real-valued function, $K_0(x)$ converges to $−ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in a branch cut, $z = 0$ is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 19:03
Thanks for the answer, Ron. I have a question related to the integral on $C_4$. Can you tell me more details how did you compute this, and is this an exact or approximate solution? My concern in this step is, as a real-valued function, $K_0(x)$ converges to $−ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in a branch cut, $z = 0$ is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 19:03
@HoangNguyen: on $C_4$, there is a ratio of two logs of very small numbers which approaches one. (Recall that we are integrating a ratio of Bessels.) The branch cut is defined by the given contour.
– Ron Gordon
Aug 8 at 19:09
@HoangNguyen: on $C_4$, there is a ratio of two logs of very small numbers which approaches one. (Recall that we are integrating a ratio of Bessels.) The branch cut is defined by the given contour.
– Ron Gordon
Aug 8 at 19:09
add a comment |Â
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Have you tried writing out the bessel functions in their Laurant series around $s=0$?
– Alex R.
Aug 8 at 17:14
Hi Alex, I did. As a real-valued function, $K_0(x)$ converges to $-ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in the principal branch cut, z = 0 is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 17:36
1
Have a look at this: math.stackexchange.com/questions/480235/…
– Ron Gordon
Aug 8 at 17:52
Thanks for the answer, Ron. I have a question related to the integral on $C_4$. Can you tell me more details how did you compute this, and is this an exact or approximate solution? My concern in this step is, as a real-valued function, $K_0(x)$ converges to $−ln(x)$, which blows up as $x rightarrow 0$. However, in the complex plane, such a function is multi-valued, and even though defined in a branch cut, $z = 0$ is still a branch point, so I doubt that an asymptotic behavior of $K_0$ around that point exists.
– Hoang Nguyen
Aug 8 at 19:03
@HoangNguyen: on $C_4$, there is a ratio of two logs of very small numbers which approaches one. (Recall that we are integrating a ratio of Bessels.) The branch cut is defined by the given contour.
– Ron Gordon
Aug 8 at 19:09