Toronto Math Forum

I mentioned about this error during lecture today

(9) Should be:
\begin{equation} u(x,t) = \frac{1}{2\sqrt{\pi kt}}e^{-\frac{x^2}{4kt}}   \end{equation}

Indeed

Error in change of variable
\begin{equation} z = x/ \sqrt{2kt} \end{equation}

(12) Shouldn't upper limit be
\begin{equation}  \frac{x}{ \sqrt{4kt}}\end{equation}


You can change variables in a different way and get different expressions; it looks like with $e^{-z^2}$ rather than $e^{-z^2/2}$ it became more standard; so I change it (but it will take time to deal with all instances in forthcoming sections, so be vigilant)

Error function shouldn't have variables in the limits
\begin{equation}erf(x) =   \sqrt{ \frac{2}{\pi} }   \int_{0}^{x} \ e^{- z^{2}/2 }dz\end{equation}

It can have variable limits or we can get constant limit but then we need integrand depending on $x$

It's actually equivalent to the wikipedia version:
\begin{equation}erf(x) = \frac{2}{ \sqrt{\pi} } \int_{0}^{x} \ e^{- t^{2} }  dt \end{equation}