# Cohen's d for unequal variance

I know my two samples have unequal variance. Should I consider this in my calculation of the effect size? I'm using Cohen's d, which Wikipedia defines as

$d = \frac{\bar{x}_1 - \bar{x}_2}{s}$

with

$s = \sqrt{\frac{(n_1-1)s^2_1 + (n_2-1)s^2_2}{n_1+n_2 - 2}}$

However, I noticed that when performing the t-test assuming unequal variance, a different estimation of the variance is used (instead of the pooled variance). Should I adapt the Cohen's d formula as well?

With unequal variances (where I find that $$\frac{S^2_{bigger}}{S^2_{smaller}} \le 1.5$$, this would be a moderate difference between variances according to Cohen, 1988) I give a range for d where I divide the difference of the means 1) by the bigger (lower bound for d) and 2) by the smaller (upper bound for d) standard deviation.