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The answers in the comments are probably the best (in particular @James S. Cook's advice to talk to a professor at a local university if you have such a connection) but here's at least a list to get you started: Given that you've got quite a range of interested there I would suggest going through the MAA's list of Math programs: https://www.maa.org/...


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… because students learning more about real numbers at university bring their understanding of real numbers that they developed through high school. Which is, real numbers are the set of rational and irrational numbers: all possible numbers on a single continuous number line. And that’s a correct working schema to begin learning more about real numbers and ...


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It's certainly beneficial for students (even if thy don't like maths) to be convinced that some manipulations are only valid under some conditions. Maybe the following example is both significant and short enough for a rapid illustration: Derivation under the integral sign is ("formally"): $$ \frac{d}{dt} \int f(x,t) dx = \int \frac{\partial f(x,t)}{\...


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Interesting integral. $$ f(w)=\frac{\cos(wx)}{1+w^2} $$ has poles at $w = \pm i$. The residue at $w=i$ is given as follows: $$ \text{Res}_{w=i}\left(\frac{\cos(wx)}{1+w^2}\right) = \text{Res}_{w=i}\left(\frac{\cos(wx)}{(w+i)(w-i)}\right) = \frac{\cos(ix)}{i+i} = \frac{\cosh(x)}{2i} $$ So, if I use the usual half-circle contour $C = [-R,R] \cup C_R^+$ where $...


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