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







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    0












    $begingroup$


    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







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      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







      share|cite|improve this question











      $endgroup$




      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






      share|cite|improve this question















      share|cite|improve this question













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      share|cite|improve this question








      edited Dec 26 '18 at 16:41







      ersh

















      asked Dec 26 '18 at 4:15









      ershersh

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          1 Answer
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          1












          $begingroup$

          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 $-10 < text{Re}(z), text{Im}(z) < 10$.



          enter image description here






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            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











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          1 Answer
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          1 Answer
          1






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          active

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          active

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          1












          $begingroup$

          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 $-10 < text{Re}(z), text{Im}(z) < 10$.



          enter image description here






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            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
















          1












          $begingroup$

          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 $-10 < text{Re}(z), text{Im}(z) < 10$.



          enter image description here






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            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














          1












          1








          1





          $begingroup$

          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 $-10 < text{Re}(z), text{Im}(z) < 10$.



          enter image description here






          share|cite|improve this answer











          $endgroup$



          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 $-10 < text{Re}(z), text{Im}(z) < 10$.



          enter image description here







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 26 '18 at 4:49

























          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


















          • $begingroup$
            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


















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