# Understanding and visualizing Bessel function with complex argument

I have a vector field in polar coordinates $$(r,\phi)$$, which I wish to visualize and better understand. It represents the Electric field vectors for a given field distribution inside a circular boundary: $$\hat{r}J_{1}[V_{01}r]+\hat{\phi}0$$, with $$V_{01}\approx 2.4$$ being the first zero of the Bessel function $$J_{0}$$. We expect such field to be radially directed. So I plotted it as follows to visualize it

TM01ErEnvelope[r_, \[Phi]_] = BesselJ[1, N[BesselJZero[0, 1]] r];

TM01Rec =
TransformedField[
"Polar" -> "Cartesian", {TM01ErEnvelope[r, \[Phi]],
0}, {r, \[Phi]} -> {x, y}]

Show[StreamPlot[TM01Rec, {x, -1, 1}, {y, -1, 1}], Graphics[Circle[]]]


This gave the correct field plot as follows

Then, following a similar line of thinking, I was faced with another (modified) system that have a difference vector field function, now with a complex argument inside the Bessel function: $$\hat{r}0+\hat{\phi}J_{1}[V_{01}r (1 - (1 + i) 0.01)]$$. So, this is a perturbed version of the argument in the previous case, with the field component $$\hat{\phi}$$ now present (instead of the $$\hat{r}$$ component). If the argument were real, one would have expected an azimuthal (concentric circular rings) vector field shape, but since the argument is complex, I am not sure how to interpret, understand or visualize this.

ModifiedTM01ErEnvelope[
r_, \[Phi]_] = BesselJ[1,
r N[BesselJZero[0, 1]] (1 - (1 + I) 0.01)]

ModifiedTM01Rec =
TransformedField[
"Polar" -> "Cartesian", {0,
ModifiedTM01ErEnvelope[r, \[Phi]]}, {r, \[Phi]} -> {x, y}]


And now I have a problem in visualizing it as a StreamPlot as I did earlier, because the result is complex.

How does one make sense of such case? How can the vector field be interpreted in such case and how can it be visualized (plotted) in a meaningful way?

The following works.

ModifiedTM01ErEnvelopeRe[r_, \[Phi]_] := Re[BesselJ[1, r N[BesselJZero[0, 1]] (1 - (1 + I) 0.01)]];
ModifiedTM01ErEnvelopeIm[r_, \[Phi]_] :=Im[BesselJ[1, r N[BesselJZero[0, 1]] (1 - (1 + I) 0.01)]];
ModifiedTM01Rec = TransformedField["Polar" -> "Cartesian",{ModifiedTM01ErEnvelopeRe[r,\[Phi],ModifiedTM01ErEnvelopeIm[r,\[Phi]]},{r,\[Phi]} -> {x, y}]
Show[StreamPlot[ModifiedTM01Rec, {x, -1, 1}, {y, -1, 1}], Graphics[Circle[]]]


Because the peturbation 1 - (1 + I) 0.01) is small, the difference between the plots is slight (but there is one).

Addition. Here is the result with $$0.7$$ instead of $$0.01$$.

• How come you replaced the $\hat{r}$ and $\hat{\phi}$ components with the Real and Imaginary components of the $\phi$ component only? The polar/cylindrical space needs $(r,\phi)$ components. In your solution you just took the components as $(\Re[f_{\phi}],\Im[f_{\phi}])$, and discarded $r$-component. Nov 28, 2020 at 21:54