I have a setup with equal sample sizes ($N_1=N_2$) from two groups (sample means $\mu_1$, $\mu_2$), but different variances $\sigma_1^2$, $\sigma_2^2$ in both groups. If the variances were equal, there are lots of well-known formulas to calculate the necessary sample size to obtain some requested statistical power.
How do I calculate the power or sample size for unequal variances? (Bonus question: Which function can I use in R?)
I am planning to test for difference of means using a Welch t-test.