Roadway and book recommendations to math study. I had some calculus, linear algebra and complex analysis courses back in college. But it is not comprehensive. And I felt that my college math was not taught in a logical sequence (maybe because my major is engineering). And some of the confusing theories may  be better appreciated if taught in more detail and in a more logical sense.

Now I had some time and not pushed by so many exams. I'd like to re-study the mathematics by myself. This is all about interest and curiosity. And I believe I can figure out all my confusions so far. But I need some guidance.

So could you guys recommend some books and logical study sequence for me? My target is as below:


(1) To fully understand the integral transform, such as fourier, laplace
and wavelet transform.
(2) To fully understand some strange fuctions, such as Dirac delta function, etc.
(3) To fully understand the Euler formula: e^ix = cos(x) + i sin(x)
(4) To fully understand the complex number "i" and complex analysis.
(5) To fully understand the differential equation.
(6) To get a thorough understanding of modern algebra.
(7) To help my work in computer science.
(8) To get the general idea of modern mathematics.


Really appreciate your advice!
I should like to point out from the beginning that several parts in your question are unclear and in some sense unanswerable (such as 'the differential equation' because there is no such thing as 'the' differential equation). Also it is unclear to me how deep you want to go into these topics so that I restrict myself to what I consider to be to the minimum. I suggest: 

(1) start with Stein's Complex analysis (Princeton lectures in analysis, vol. II), chapter 1 (this will answer part 2 and 3 of your question, apart from the complex analysis part, for which you also have to look at least at chapters 2-4). 

(2) Part 1 of your question will be answered by Stein's Fourier analysis (same series, vol. I), chap. 5 and 6. The fourier transform comes up in increasing level of sophistication in Stein's series, such as in chap. 2 of vol. II and chap. 5. of vol. III. A good reference for integral transforms is Antimirov's applied integral transforms, chap. 2.

(3) Part 5: the basic material can be found in Arnold's ODE text chap. 1-4 (several sections could be omitted). But of course one could endlessly read about differential equations; differential equations come up also in several place of Stein's series (so I suggest you get these books, they are relatively expensive but they are excellent in any aspect, and also suitably elementary).

(4) Part 2: frankly speaking there is no such thing as the Dirac delta function, the road leads to distribution theory, which is dealt with in chapter 3 of Stein's functional analysis (vol. IV of the series). If you want to know something about special (gamma, zeta, elliptic and theta) functions, a good place to look is Stein's complex analysis, chap. 6-7, 9-10.

(5) Part 6: Lang's algebra book (GTM), as much as you can, at any rate chapters 1-6 (mainly group, ring, module and field/galois theory).

(6) Part 7-8: This is difficult to answer, but the more mathematics you understand the better your thinking. As far as 8 is concerned, Gower's Princeton companion to mathematics gives you a good overview of what mathematics is about. But again this is a very broad question; however if you get to know some category theory (Mac Lane's book), you will certainly understand many ideas underlying the algebra parts of mathematics.

By the way there is not much to be said about Euler's formula, some people find it special but it is in fact a triviality.
Functional Analysis - Kreyszig

Methods of Mathematical Physics- Courant &amp; Hilbert or Arfken

Complex Analysis - Stein &amp; Shakarchi or Rudin or Ahlfors

Differential Equations - G.F. Simmons

Algebra- Herstein and thereafter Lang/ Hungerford/ Jacobson

Concrete Mathematics - Ronald L. Graham, Knuth, Patashnik

Hope this is helpful.