What would be a quick way to learn these topics in Applied Mathematics? 
UPDATE: So it seems like the more realistic option would be to aim for it next term. That would give me a couple of months to study and I would have to go for the full syllabus. Assuming my prior knowledge is close to nil, what would be the route you guys suggest I take.

I am pursuing an Information Technology degree and I have hit a roadblock which is not allowing me to go further. During high school, I had chosen the lowest level mathematics I could take because of which I skipped a lot of topics that seem to be important now. 
I cannot continue further with my degree until I learn and pass Applied Mathematics II, as it is not an optional subject at the University of Mumbai, in which I am enrolled. The professor told me that there is an exam I can sit for in three days which will mainly test the following topics -   
De Moivre's Theorem, 
Logarithms of complex numbers,
Cauchy Riemann equations,
Properties of Laplace Transform,
Laplace of standard function,
Error functions,
Fourier series,
Double integrals
Either I pass the exam with these topics or I have to take an exam with  the full syllabus later. Whichever way I look at it, this exam is something I need to pass. At this point I don't care about what grade I get in the subject, just that I manage to gain a good enough understanding to be able pass the exam.
I need your help to be able to figure out a good way to be able to approach the topics and in terms of which online resources I should look at using for my study.
Note: I also think it's important that I point out that because I took the lowest level math in high school I did not learn differentiation, limits and integration.
Note: The link for the syllabus also provides some recommended references, but honestly I don't know how good they will be because my textbook does a miserable job of breaking down the topics and gives no understanding of the perquisites needed.
 A: To be honest, your odds of passing the exam that is in 3 days is not good. Since you do not know "differentiation, limits and integration", you won't be able to grasp any of the topics actually tested. So unless more than 50% of the exam is going to be regurgitating definitions, you won't pass.
Also, even if you take it next term, you have a lot of catching up you need to do.
A: In 3 days? Relax for now to consider doing it during next term. There are the Khan Acadamy, YouTube etc. classes, but no substitute for learning by your own problem solving practice. Remember, you can excel on a solid foundation that time gives. 
A: Good references are William Chen's lecture notes. They cover from entry-level college mathematics to rather advanced topics, quite well written. For your specific case, check out "First year calculus", "Introduction to complex analysis", and perhaps "Discrete mathematics". Take a look at syllabi, check out the books/lecture notes referenced.
Try to solve problems by yourself first, then peek at solutions or ask here (without your previous tries, you won't understand solutions, not really). Take a look at how to write mathematics (Knuth, Larrabee, Roberts "Mathematical Writing" is a nice introduction) understand how to write proofs (Hammack's "Book of Proof" is a useful reference). How to express yourself clearly and precisely, how to lay out an argument is critical in mathematics. And in programming, and any technical field whatsoever.
Don't rush it. It is better to finish a few terms late than not finish at all.
