Complex zeroes of Error Function and Parabolic Cylinder Function.

Question

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.

Answer 1

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

Answer 2

You need the inverse function given with Bessel Y Zero $\text y_{v,x}$:

$$\frac{\text D_{-\frac12}\left(iy\right)}{\text D_{-\frac12}\left(y\right)}=-e^{2ix}\implies y=(1-i)\sqrt{2\text y_{\frac14,\Bbb N-\frac{\tan^{-1}(\tan(x))}\pi}}, x\in\Bbb R $$

which works with the $n$th natural number giving the $n$th solution

Another presentation is:

$$\frac{\text D_{-\frac12}\left(iy\right)}{\text D_{-\frac12}\left(y\right)}= \frac{x+i}{x-i}\implies y=(1-i)\sqrt{2\text y_{\frac14,\frac{\tan^{-1}(x)}\pi+\Bbb N}}, x\in\Bbb R $$

which also works with the $n$th solution from the $n$ natural number.

Both of these solutions partly solve your $r(z)=a\operatorname D_{-\frac12}(z)+b\operatorname D_{-\frac12}(iz)$