It's nice to see a non-typophile (or maybe you are one by now :-) pay such attention to typography.

Since Amerigo is rather high-contrast (or at least that the thinnest parts are very thin) its suitability will depend on the point size and most of all the reproduction method. For example at 12 point on a laser printer it might be great. One thing going against Amerigo though is that it has flare serifs, not full serifs. In a PhD thesis high readability is important (and most people agree -with good reason- that serifs help).

But as you seem to realize it's Euler that's going to be calling the shots* in terms of what the other font should be, and to me Amerigo doesn't cut it. Your safest bet is Aldus/Palatino. Those two are basically sisters: the former is for text, the latter for display - and you might try using both, with Palatino for mixing within equations (as they've done with "cos" and "sin" there - although I wonder why they didn't simply use Euler for those). But this is all too predictable...

* Unless you can use something else - something more sober, less cursive.

If you're feeling up to the task you might instead look for something different that has everything you need - and it's possible to find serif fonts that have a subtly handwritten feel. What I can tell you though is that your biggest challenge will be getting the weights to harmonize: especially if the repro is high-quality the weights would have to be quite close to avoid a jarring effect.

hhp

Thank you for your answer. My Idea was to use a main body of 10pt, so you suggest to avoid the Amerigo at that size.

Unfortunately the set of possible fonts with math support is somewhat limited (http://www.tug.dk/FontCatalogue/mathfonts.html). A suitable selection of fonts is:

Antykwa family, but I think it is too fancy for a long text:

Computer Concrete, that I really like:

Artemisia gives the idea of a handwritten serif font:

Always nice Garamond:

Palatino:

And finally the Times, that I really can not stand:

Please, dont't use Copperplate Gothic! ;-)