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i'm a computer science student and i'm trying to analytically find the value of the convolution between an ideal step-edge and either a gaussian function or a first order derivative of a gaussian function. In other words, given an ideal step edge with amplitude $A$ and offset $B$: $$ i(t)=\left\{ \begin{array}{l l} A+B & \quad \text{if $t \ge t_{0}$}\\ B & \quad \text{if $t \lt t_{0}$}\\ \end{array} \right. $$ and the gaussian function and it's first order derivative $$ g(t) = \frac{1}{\sigma \sqrt{2\pi}}e^{- \frac{(t - \mu)^2}{2 \sigma^2}}\\ g'(t) = -\frac{t-\mu}{\sigma^3 \sqrt{2\pi}}e^{- \frac{(t - \mu)^2}{2 \sigma^2}} $$ i'd like to calculate the value of both $$ o(t) = i(t) \star g(t)\\ o'(t) = i(t) \star g'(t) $$ at time $t_{0}$ ( i.e. $o(t_{0})$ and $o'(t_{0}) )$. I tried to solve the convolution integral but unfortunately i'm not so matematically skilled to do it. Can you help me? Thank you in advance very much.

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