Complex analysis or calculus?

[Neutrin0]

Im currently reading mathematics for physicists by Philippe Dennery and André Krzywicki, and I’m understanding most concepts however I think it would be better for me to get a book on complex analysis or calculus to better understand it so I’m not left looking at an equation for an hour trying to figure out what it means. So here comes the split, do I get a complex analysis book? Or a calculus book? I might be able to Borrow a calculus textbook from my math teacher study that for a bit and then move onto the next book, which would be the complex analysis. And I’m pretty sure the book mathematics for physicists has some linear algebra to so I might get a linear algebra book to. But I’d rather get an answer from experts then assume something.

 

[MidgetDwarf]

CAlculus is typically learned before complex...

So calculus.

 

[chth]

If you haven't mastered differential, integral and multivariate calculus at the freshman level, then that should be your focus. If you have, then the typical freshman textbooks in calculus won't help you much. Maybe a more advanced text in real analysis, differential equations or a similar subject would help you understand the physics. Regardless, if your background in calculus and real analysis is not sufficient, complex analysis will seem impossibly difficult.

 

[Neutrin0]

If you haven't mastered differential, integral and multivariate calculus at the freshman level, then that should be your focus. If you have, then the typical freshman textbooks in calculus won't help you much. Maybe a more advanced text in real analysis, differential equations or a similar subject would help you understand the physics. Regardless, if your background in calculus and real analysis is not sufficient, complex analysis will seem impossibly difficult.

Thank you for the advice, I did take a look at my schools calc book and if I’m not mistaken it did have some of the subjects you mentioned. And if it doesn’t there are online sources that I can use in case, regardless I’ll make sure to focus on what you said first and then move onto complex analysis.

 

[Muu9]

A lot of complex analysis books are super rigorous and meant for mathematicians rather than physics students. A good applied text is the one by Zill - the first chapter starts with complex numbers so you can do it before calculus, but it will eventually start using calculus and assume you already learned it.

 

[WWGD]

A lot of complex analysis books are super rigorous and meant for mathematicians rather than physics students. A good applied text is the one by Zill - the first chapter starts with complex numbers so you can do it before calculus, but it will eventually start using calculus and assume you already learned it.

Schaum's Calculus too, is a good applied text. An underrated collection, imo.

 

[Muu9]

Schaum's Calculus too, is a good applied text. An underrated collection, imo.

You mean this one?

 

[WWGD]

You mean this one?

Yes, that one. Could complement other textbooks, and there are cheap used copies of it.

 

[DEvens]

You mean this one?

That's for complex numbers. This one for calculus.

https://www.amazon.com/Schaums-Solved-Problems-Calculus-Outlines/dp/0071635343/?tag=pfamazon01-20

And if that one is too easy, this one.

https://www.amazon.com/Schaums-Outline-Advanced-Calculus-Outlines/dp/0071623663/?tag=pfamazon01-20

Schaum's are pretty good for a large array of topics. They are a quite good learning aid if you like solved problems. And they are comparatively cheap. Most students have limited money. They also have a couple handbooks that are potential to keep close by for many years. Integral tables and Laplace transforms and math formulas and a few others. So if you are doing hand calculation or numerical computer calcs, browse through the Schaum's catalog. Cheaper than the CRC handbook.

 

[Muu9]

That's for complex numbers

The book I linked also covers calculus in the complex plane.

 

[gmax137]

Schaum's are pretty good for a large array of topics. They are a quite good learning aid if you like solved problems. And they are comparatively cheap.

I have always liked learning by example. Here's my Schaums collection. Some I bought new, some used, some given to me by retiring coworkers.

 

[MidgetDwarf]

I found schaums useful for things like probability [Markov Chains], PDE, Toplogy, complex analysis Ie., for more examples to apply the theory learned from pure mathbooks.

 

[Muu9]

I have always liked learning by example. Here's my Schaums collection. Some I bought new, some used, some given to me by retiring coworkers.

View attachment 367957

Did you do your bachelor's in engineering?

 

[gmax137]

Did you do your bachelor's in engineering?

No, physics bachelor's; nuclear eng masters.

 

[mathwonk]

real Calculus comes before complex calculus. And when learning real calculus, the topics that will be especially useful in complex calculus are power series and path integration. It helps also to focus on the actual definition of the Riemann integral as approximated by sums. We have a bad tendency in real calculus of ignoring the actual meaning of the integral, and just reaching immediately for an antiderivative, i.e. relying too much on the Fundamental Theorem of Calculus. This only works because in real calculus we are always working on intervals, which are contractible sets. In complex calculus we do frequently work on contractible open sets like discs, but we also work on more complicated ones like punctured discs. In those sets, the FTC is often not available, i.e. antiderivatives often do not exist globally, (e.g. for 1/z in the punctured unit disc) and you have to rely on the actual definition of an integral, not just finding an antiderivative. Other topics that do not arise in real calculus focus on the geometry of sets in the plane and the extended plane, "conformal mappings". A key feature of complex analysis arising from the fact that all complex differentiable functions are "analytic", i.e. have local power series representations, is the principle of analytical continuation, i.e. that an analytic function is determined along any path just by its values near the initial point, and the open mapping principle, the image of an open set is always again open. So you must learn some topology, open sets, closed sets, and connectedness and simple connectedness. One bonus is that almost all books on complex analysis are good. The subject is so beautiful and coherent it is apparently hard to explain it badly. Churchill is a classic, my favorite beginner's book is Greenleaf, and favorite (but succinct) theoretical book is Cartan. Lang is also good. Well they are all pretty good, if in different ways. The last chapter of Courant's classic real calculus text, vol.2, is a brief but clear introduction to complex calculus.

 

[mathwonk]

The good news is: once one appreciates the need for a Riemann sum type definition of a complex path integral, it is possible then to use antiderivatives to evaluate them. I.e. once a path is parametrized, one pulls the integral back to the real parameter interval, and uses the usual real calculus FTC. And for comparing integrals over different paths, one adapts the Green's theorem from 2 variable real calculus, to get Cauchy's theorem. The Cauchy Riemann differential equations come in here.
As for integrating functions like 1/z around the unit circle, the parametrization technique works, and it turns out this is actually the only analytic function in a punctured disc that does not have an antiderivative. I.e. an analytic function f(z) in the punctured unit disc always has form g(z) + c/z, where g does have an antiderivative in the whole punctured disc. hence once you have integrated 1/z you can integrate them all, and this is called the residue theorem. beautiful stuff.

Oh yes, the contour lines of analytic functions, i.e. of their real and imaginary parts, (which are orthogonal because of the C-R diff. equation), called harmonic functions, mimic physical phenomena like electrical field and fluid flow lines. Indeed these insights seem to have led the 19th century mathematicians to some their best results.

hence complex calculus is a sort of very nice special case, and refinement, of real calculus of two variables, so one needs a good familiarity with not only one variable real calculus, but of 2 variable real calculus also.