# Toronto Math Forum

## APM346-2015F => APM346--Misc => Textbook errors => Topic started by: Bruce Wu on October 07, 2015, 07:01:08 PM

Title: Section 3.1 error in equation 9
Post by: Bruce Wu on October 07, 2015, 07:01:08 PM
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter3/S3.1.html#mjx-eqn-eq-3.1.9

The $kt$ in the denominator should also be under the square root. Similar error in the line below as a part of remark 2.
Title: Re: Section 3.1 error in equation 9
Post by: Andrew Lee Chung on October 07, 2015, 11:24:01 PM

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

Indeed

Error in change of variable
$$z = x/ \sqrt{2kt}$$

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

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
$$erf(x) = \sqrt{ \frac{2}{\pi} } \int_{0}^{x} \ e^{- z^{2}/2 }dz$$

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:
$$erf(x) = \frac{2}{ \sqrt{\pi} } \int_{0}^{x} \ e^{- t^{2} } dt$$