Complex zeroes of Error Function and Parabolic Cylinder Function.
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Does there exist EXACT zeros of Error Function (Erfc(z)) and Parabolic Cylinder Function ($D_v(z)$)(http://functions.wolfram.com/HypergeometricFunctions/ParabolicCylinderD/). Here (https://dlmf.nist.gov/12) (section 12.2 and 12.11) is the discussion about the real zeros of $D_v(z)$ but I don't understand about the complex zeros. I am particularly interested for the exact complex zeros of $D_{-frac{1}{2}}(z)$ and $D_{-frac{1}{2}}(iz)$.
I think I would be able to proceed on my work, if equivalently, I know if there exists a zero (except $z=0$) of error function.
complex-analysis special-functions mathematical-physics
asked Dec 26 '18 at 4:15
ershersh
357113
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add a comment |
$begingroup$
Does there exist EXACT zeros of Error Function (Erfc(z)) and Parabolic Cylinder Function ($D_v(z)$)(http://functions.wolfram.com/HypergeometricFunctions/ParabolicCylinderD/). Here (https://dlmf.nist.gov/12) (section 12.2 and 12.11) is the discussion about the real zeros of $D_v(z)$ but I don't understand about the complex zeros. I am particularly interested for the exact complex zeros of $D_{-frac{1}{2}}(z)$ and $D_{-frac{1}{2}}(iz)$.
I think I would be able to proceed on my work, if equivalently, I know if there exists a zero (except $z=0$) of error function.
complex-analysis special-functions mathematical-physics
asked Dec 26 '18 at 4:15
ershersh
357113
$endgroup$
add a comment |
$begingroup$
Does there exist EXACT zeros of Error Function (Erfc(z)) and Parabolic Cylinder Function ($D_v(z)$)(http://functions.wolfram.com/HypergeometricFunctions/ParabolicCylinderD/). Here (https://dlmf.nist.gov/12) (section 12.2 and 12.11) is the discussion about the real zeros of $D_v(z)$ but I don't understand about the complex zeros. I am particularly interested for the exact complex zeros of $D_{-frac{1}{2}}(z)$ and $D_{-frac{1}{2}}(iz)$.
I think I would be able to proceed on my work, if equivalently, I know if there exists a zero (except $z=0$) of error function.
complex-analysis special-functions mathematical-physics
asked Dec 26 '18 at 4:15
ershersh
357113
$endgroup$
Does there exist EXACT zeros of Error Function (Erfc(z)) and Parabolic Cylinder Function ($D_v(z)$)(http://functions.wolfram.com/HypergeometricFunctions/ParabolicCylinderD/). Here (https://dlmf.nist.gov/12) (section 12.2 and 12.11) is the discussion about the real zeros of $D_v(z)$ but I don't understand about the complex zeros. I am particularly interested for the exact complex zeros of $D_{-frac{1}{2}}(z)$ and $D_{-frac{1}{2}}(iz)$.
I think I would be able to proceed on my work, if equivalently, I know if there exists a zero (except $z=0$) of error function.
complex-analysis special-functions mathematical-physics
complex-analysis special-functions mathematical-physics
asked Dec 26 '18 at 4:15
ershersh
357113
asked Dec 26 '18 at 4:15
ershersh
357113
edited Dec 26 '18 at 16:41
ersh
asked Dec 26 '18 at 4:15
ershersh
357113
asked Dec 26 '18 at 4:15
ershersh
357113
asked Dec 26 '18 at 4:15
ershersh
357113
357113
add a comment |
add a comment |
1 Answer
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Both erfc and $D_{-1/2}$ have infinitely many complex zeros. I doubt very much that you'll get closed forms for them. The closest zeros to $0$ for $text{erfc}$ are approximately
$-1.354810128 pm 1.991466843 i$. The closest zeros to $0$ for $text{D}_{-1/2}$ are approximately $-2.110851502 pm 2.267049242 i$.
Here are the zeros of erfc (red) and $D_{-1/2}$ (blue) for $-10 < text{Re}(z), text{Im}(z) < 10$.
answered Dec 26 '18 at 4:37
Robert IsraelRobert Israel
323k23212466
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$begingroup$
Just existential proof of such zeros for $D_{-frac{1}{2}}$ would work for me! Please share if you know the references. Furthermore, I am precisely looking for at least one non-zero zero (with nonzero imaginary part) of function $r(z)= a D_{-frac{1}{2}}(z) + b D_{-frac{1}{2}}(iz)$ where $a$ and $b$ are constants.
$endgroup$
– ersh
Dec 26 '18 at 4:49
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If you can approximate $D_{-1/2}$ (or any other analytic function $f$) numerically to a given accuracy, you can prove existence of a zero in a region bounded by a curve by integrating $f'/f$ around the curve.
$endgroup$
– Robert Israel
Dec 26 '18 at 4:51
$begingroup$
I don't know any good references on this stuff. Do't understand how to proceed. Could you please be more specific like how to approximate $D_{-frac{1}{2}}$?
$endgroup$
– ersh
Dec 26 '18 at 15:20
$begingroup$
There is a representation of $D_v(z)$ through confluent hypergeometric function (functions.wolfram.com/HypergeometricFunctions/…). I think there is some on the zeros of confluent hypergeometric function (e.g; here ac.els-cdn.com/0021904582900764/…). In this paper at page 5, the bounds for zeros are given for real parameters $a$ and $c$. If the parameters are real in confluent hypergeo function, does its zeros are always real?
$endgroup$
– ersh
Dec 26 '18 at 16:37
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Both erfc and $D_{-1/2}$ have infinitely many complex zeros. I doubt very much that you'll get closed forms for them. The closest zeros to $0$ for $text{erfc}$ are approximately
$-1.354810128 pm 1.991466843 i$. The closest zeros to $0$ for $text{D}_{-1/2}$ are approximately $-2.110851502 pm 2.267049242 i$.
Here are the zeros of erfc (red) and $D_{-1/2}$ (blue) for $-10 < text{Re}(z), text{Im}(z) < 10$.
answered Dec 26 '18 at 4:37
Robert IsraelRobert Israel
323k23212466
$endgroup$
$begingroup$
Just existential proof of such zeros for $D_{-frac{1}{2}}$ would work for me! Please share if you know the references. Furthermore, I am precisely looking for at least one non-zero zero (with nonzero imaginary part) of function $r(z)= a D_{-frac{1}{2}}(z) + b D_{-frac{1}{2}}(iz)$ where $a$ and $b$ are constants.
$endgroup$
– ersh
Dec 26 '18 at 4:49
$begingroup$
If you can approximate $D_{-1/2}$ (or any other analytic function $f$) numerically to a given accuracy, you can prove existence of a zero in a region bounded by a curve by integrating $f'/f$ around the curve.
$endgroup$
– Robert Israel
Dec 26 '18 at 4:51
$begingroup$
I don't know any good references on this stuff. Do't understand how to proceed. Could you please be more specific like how to approximate $D_{-frac{1}{2}}$?
$endgroup$
– ersh
Dec 26 '18 at 15:20
$begingroup$
There is a representation of $D_v(z)$ through confluent hypergeometric function (functions.wolfram.com/HypergeometricFunctions/…). I think there is some on the zeros of confluent hypergeometric function (e.g; here ac.els-cdn.com/0021904582900764/…). In this paper at page 5, the bounds for zeros are given for real parameters $a$ and $c$. If the parameters are real in confluent hypergeo function, does its zeros are always real?
$endgroup$
– ersh
Dec 26 '18 at 16:37
add a comment |
$begingroup$
Both erfc and $D_{-1/2}$ have infinitely many complex zeros. I doubt very much that you'll get closed forms for them. The closest zeros to $0$ for $text{erfc}$ are approximately
$-1.354810128 pm 1.991466843 i$. The closest zeros to $0$ for $text{D}_{-1/2}$ are approximately $-2.110851502 pm 2.267049242 i$.
Here are the zeros of erfc (red) and $D_{-1/2}$ (blue) for $-10 < text{Re}(z), text{Im}(z) < 10$.
answered Dec 26 '18 at 4:37
Robert IsraelRobert Israel
323k23212466
$endgroup$
$begingroup$
Just existential proof of such zeros for $D_{-frac{1}{2}}$ would work for me! Please share if you know the references. Furthermore, I am precisely looking for at least one non-zero zero (with nonzero imaginary part) of function $r(z)= a D_{-frac{1}{2}}(z) + b D_{-frac{1}{2}}(iz)$ where $a$ and $b$ are constants.
$endgroup$
– ersh
Dec 26 '18 at 4:49
$begingroup$
If you can approximate $D_{-1/2}$ (or any other analytic function $f$) numerically to a given accuracy, you can prove existence of a zero in a region bounded by a curve by integrating $f'/f$ around the curve.
$endgroup$
– Robert Israel
Dec 26 '18 at 4:51
$begingroup$
I don't know any good references on this stuff. Do't understand how to proceed. Could you please be more specific like how to approximate $D_{-frac{1}{2}}$?
$endgroup$
– ersh
Dec 26 '18 at 15:20
$begingroup$
There is a representation of $D_v(z)$ through confluent hypergeometric function (functions.wolfram.com/HypergeometricFunctions/…). I think there is some on the zeros of confluent hypergeometric function (e.g; here ac.els-cdn.com/0021904582900764/…). In this paper at page 5, the bounds for zeros are given for real parameters $a$ and $c$. If the parameters are real in confluent hypergeo function, does its zeros are always real?
$endgroup$
– ersh
Dec 26 '18 at 16:37
add a comment |
$begingroup$
Both erfc and $D_{-1/2}$ have infinitely many complex zeros. I doubt very much that you'll get closed forms for them. The closest zeros to $0$ for $text{erfc}$ are approximately
$-1.354810128 pm 1.991466843 i$. The closest zeros to $0$ for $text{D}_{-1/2}$ are approximately $-2.110851502 pm 2.267049242 i$.
Here are the zeros of erfc (red) and $D_{-1/2}$ (blue) for $-10 < text{Re}(z), text{Im}(z) < 10$.
answered Dec 26 '18 at 4:37
Robert IsraelRobert Israel
323k23212466
$endgroup$
Both erfc and $D_{-1/2}$ have infinitely many complex zeros. I doubt very much that you'll get closed forms for them. The closest zeros to $0$ for $text{erfc}$ are approximately
$-1.354810128 pm 1.991466843 i$. The closest zeros to $0$ for $text{D}_{-1/2}$ are approximately $-2.110851502 pm 2.267049242 i$.
Here are the zeros of erfc (red) and $D_{-1/2}$ (blue) for $-10 < text{Re}(z), text{Im}(z) < 10$.
answered Dec 26 '18 at 4:37
Robert IsraelRobert Israel
323k23212466
edited Dec 26 '18 at 4:49
answered Dec 26 '18 at 4:37
Robert IsraelRobert Israel
323k23212466
answered Dec 26 '18 at 4:37
Robert IsraelRobert Israel
323k23212466
answered Dec 26 '18 at 4:37
Robert IsraelRobert Israel
323k23212466
323k23212466
$begingroup$
Just existential proof of such zeros for $D_{-frac{1}{2}}$ would work for me! Please share if you know the references. Furthermore, I am precisely looking for at least one non-zero zero (with nonzero imaginary part) of function $r(z)= a D_{-frac{1}{2}}(z) + b D_{-frac{1}{2}}(iz)$ where $a$ and $b$ are constants.
$endgroup$
– ersh
Dec 26 '18 at 4:49
$begingroup$
If you can approximate $D_{-1/2}$ (or any other analytic function $f$) numerically to a given accuracy, you can prove existence of a zero in a region bounded by a curve by integrating $f'/f$ around the curve.
$endgroup$
– Robert Israel
Dec 26 '18 at 4:51
$begingroup$
I don't know any good references on this stuff. Do't understand how to proceed. Could you please be more specific like how to approximate $D_{-frac{1}{2}}$?
$endgroup$
– ersh
Dec 26 '18 at 15:20
$begingroup$
There is a representation of $D_v(z)$ through confluent hypergeometric function (functions.wolfram.com/HypergeometricFunctions/…). I think there is some on the zeros of confluent hypergeometric function (e.g; here ac.els-cdn.com/0021904582900764/…). In this paper at page 5, the bounds for zeros are given for real parameters $a$ and $c$. If the parameters are real in confluent hypergeo function, does its zeros are always real?
$endgroup$
– ersh
Dec 26 '18 at 16:37
add a comment |
$begingroup$
Just existential proof of such zeros for $D_{-frac{1}{2}}$ would work for me! Please share if you know the references. Furthermore, I am precisely looking for at least one non-zero zero (with nonzero imaginary part) of function $r(z)= a D_{-frac{1}{2}}(z) + b D_{-frac{1}{2}}(iz)$ where $a$ and $b$ are constants.
$endgroup$
– ersh
Dec 26 '18 at 4:49
$begingroup$
If you can approximate $D_{-1/2}$ (or any other analytic function $f$) numerically to a given accuracy, you can prove existence of a zero in a region bounded by a curve by integrating $f'/f$ around the curve.
$endgroup$
– Robert Israel
Dec 26 '18 at 4:51
$begingroup$
I don't know any good references on this stuff. Do't understand how to proceed. Could you please be more specific like how to approximate $D_{-frac{1}{2}}$?
$endgroup$
– ersh
Dec 26 '18 at 15:20
$begingroup$
There is a representation of $D_v(z)$ through confluent hypergeometric function (functions.wolfram.com/HypergeometricFunctions/…). I think there is some on the zeros of confluent hypergeometric function (e.g; here ac.els-cdn.com/0021904582900764/…). In this paper at page 5, the bounds for zeros are given for real parameters $a$ and $c$. If the parameters are real in confluent hypergeo function, does its zeros are always real?
$endgroup$
– ersh
Dec 26 '18 at 16:37
$begingroup$
Just existential proof of such zeros for $D_{-frac{1}{2}}$ would work for me! Please share if you know the references. Furthermore, I am precisely looking for at least one non-zero zero (with nonzero imaginary part) of function $r(z)= a D_{-frac{1}{2}}(z) + b D_{-frac{1}{2}}(iz)$ where $a$ and $b$ are constants.
$endgroup$
– ersh
Dec 26 '18 at 4:49
$begingroup$
Just existential proof of such zeros for $D_{-frac{1}{2}}$ would work for me! Please share if you know the references. Furthermore, I am precisely looking for at least one non-zero zero (with nonzero imaginary part) of function $r(z)= a D_{-frac{1}{2}}(z) + b D_{-frac{1}{2}}(iz)$ where $a$ and $b$ are constants.
$endgroup$
– ersh
Dec 26 '18 at 4:49
$begingroup$
If you can approximate $D_{-1/2}$ (or any other analytic function $f$) numerically to a given accuracy, you can prove existence of a zero in a region bounded by a curve by integrating $f'/f$ around the curve.
$endgroup$
– Robert Israel
Dec 26 '18 at 4:51
$begingroup$
If you can approximate $D_{-1/2}$ (or any other analytic function $f$) numerically to a given accuracy, you can prove existence of a zero in a region bounded by a curve by integrating $f'/f$ around the curve.
$endgroup$
– Robert Israel
Dec 26 '18 at 4:51
$begingroup$
I don't know any good references on this stuff. Do't understand how to proceed. Could you please be more specific like how to approximate $D_{-frac{1}{2}}$?
$endgroup$
– ersh
Dec 26 '18 at 15:20
$begingroup$
I don't know any good references on this stuff. Do't understand how to proceed. Could you please be more specific like how to approximate $D_{-frac{1}{2}}$?
$endgroup$
– ersh
Dec 26 '18 at 15:20
$begingroup$
There is a representation of $D_v(z)$ through confluent hypergeometric function (functions.wolfram.com/HypergeometricFunctions/…). I think there is some on the zeros of confluent hypergeometric function (e.g; here ac.els-cdn.com/0021904582900764/…). In this paper at page 5, the bounds for zeros are given for real parameters $a$ and $c$. If the parameters are real in confluent hypergeo function, does its zeros are always real?
$endgroup$
– ersh
Dec 26 '18 at 16:37
$begingroup$
There is a representation of $D_v(z)$ through confluent hypergeometric function (functions.wolfram.com/HypergeometricFunctions/…). I think there is some on the zeros of confluent hypergeometric function (e.g; here ac.els-cdn.com/0021904582900764/…). In this paper at page 5, the bounds for zeros are given for real parameters $a$ and $c$. If the parameters are real in confluent hypergeo function, does its zeros are always real?
$endgroup$
– ersh
Dec 26 '18 at 16:37
add a comment |
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