There is no simple representation of such formula. You are likely looking for a simple one. You can find explanation about kde-estimations at Wikipedia. So, kde-estimation is a function defined as a sequence of calculations (these calculations are hard to perform manually, but easy to do with help of the computer), it is a computational procedure.
If you want to get a "convenient" formula as an estimation of pdf, you can look at parametric probability density estimations, e.g. fit a normal distribution, or fit a mixture of normal distributions (using mixture of known distributions might be very flexible, if you want to handle non-normally distributed data).
If you want to get a "convenient" formula as an estimation of pdf, you can look at parametric probability density estimations, e.g. fit a normal distribution, or fit a mixture of normal distributions (using mixture of known distributions might be very flexible, if you want to handle non-normally distributed data).