Error function property

Question

I have a question regarding a property of the error function.

Is $k\cdot\text{erfc}(-x) = 1-k\cdot\text{erfc}(x)$ for all real $x$ for any $k$?

Answer 1

Obviously this cannot be true, because for $k=0$ this would yield $0=1$.

Your comment suggest however that you are interested in the equality: $$ 1-\text{erfc}(x) = \frac{1}{2}\left(\text{erfc}(-x)-\text{erfc}(x)\right) $$ This is indeed true, because $$1-\text{erfc}(x) = \text{erf}(x) = \frac{1}{2}\left(\text{erf}(x)-\text{erf}(-x))\right) = \frac{1}{2}\left(\text{erfc}(-x)-\text{erfc}(x)\right)$$

Here I made use of the definition of $\text{erfc}$ and the anti-symmetry of $\text{erf}$.