Importance of Complex Analysis

[heman]

Hi PPls
okay i have studied calculus and i can easily see its application in many things like calculating volume,areas,rates ..etc. but i want to know what is the application of complex analysis...where does it all find its uses and why one study it??

 

[ahrkron]

Just to mention a couple: Quantum Mechanics *needs* complex numbers (and hence good math tools for complex functions), and electrical engineering makes extensive use of c. numbers to simplify calculations.

 

[quantumdude]

"The shortest path between two truths in the real domain passes through the complex domain."
--J. Hadamard

Here's one of my favorite examples. I taught it to my Electric Circuits course last summer, and it blew their little minds.

When they learned to take the Laplace Transform in Differential Equations they had to do the transforms of sin(x) and cos(x) seperately, and both of them involved a double integration by parts. But if instead you use the Euler identity $e^{ix}=cos(x)+isin(x)$ then you get:

$$L[e^{iat}]=\int_0^\infty e^{iat}e^{-st}dt=\int_0^\infty e^{(-s+ia)t}$$
$$L[e^{iat}]=\frac{-1}{s-ia}\lim_{k_\rightarrow\infty}[e^{(-s+ia)k}-1]$$
$$L[e^{iat}]=\frac{-1}{s-ia}(-1)=\frac{s+ia}{s^2+a^2}$$

This integral is not only easier than the trigonometric integrals I described earlier, it also gives you 2 transforms for the price of 1. Take the real part and you have $L[cos(at)]$, and take the imaginary party and you have $L[sin(at)]$.

And that's just using a complex valued function of a real variable. It doesn't even begin to tap the power of functions of complex variables.

 

[arildno]

Cool&applicable things from complex analysis:
Conformal mapping, method of stationary phase, method of steepest descent, and lots of other cool stuff..

 

[heman]

Just to mention a couple: Quantum Mechanics *needs* complex numbers (and hence good math tools for complex functions), and electrical engineering makes extensive use of c. numbers to simplify calculations.

quantum mechanics is bit advanced...
eletical engineering ...how does it makes use of C
what can be its applications in everday life
Thx

 

[dextercioby]

Think about the theory of special functions.Starting with simple exponential and ending with parabolic cylindrical functions,Whittaker functions,Gauss hyper geom.functions,...

They're everywhere.From the simplest integral to the most devious system of PDE-s.

Daniel.

 

[mathwonk]

i like it because it gets rid of the sin and cosine, so there is no such thing as trig, just exponentials.

i.e. sin(x) = (1/2i)[e^(ix) - e^(-ix)].

this was actually the definition given in my freshman calc class, where e^z was defiend by a convergent power series.


now i like it because it makes geometry easier: i.e. algebraic subsets of complex projective space always intersect when they should, polyno,mials always have the right numnber of roots. so it just opens your eyes to what is hidden in real calculus.

 

[Curious3141]

quantum mechanics is bit advanced...
eletical engineering ...how does it makes use of C
what can be its applications in everday life
Thx

A.C. circuit analysis can be made a lot easier by representing phasors as complex numbers. Impedances become real, imaginary or complex. For instance, the impedance of a pure inductance and pure capacitance are [itex]j\omega L, \frac{1}{j\omega C}[/tex] respectively. Look up any intermediate to advanced circuit theory book for better theory treatment and worked examples.

 

[arildno]

heman:
As others, like mathwonk, have mentioned (or implied) exponential functions and trigonometric functions are essentially the "same" in complex analysis!

Effectively, the study of exponential decay/growth and oscillations becomes immensely simplified, in particular where you have both behaviours present.
This will most likely be your first encounter with applications of complex analysis.

 

[goldi]

I want to hear urs comments on this:
How is Cauchy's theorem important in Complex Analysis...

 

[arildno]

I want to hear urs comments on this:
How is Cauchy's theorem important in Complex Analysis...

About as valuable as a crown jewel.

 

[mathwonk]

that theorem implies that every complex function with one derivative throughout a region has actually infinitely many derivatives, and even equals its own taylor series locally everywhere.

cauchys integral theorem is the compelx analog of stokes theorem which we have discussed many times here has many applications ni topology and elsewhere.

it implies the residue theorem the argument principle, etc etc, essentially everything in basic complex analysis.

I have wasted a few minutes searching no the web for an adequate statement of the theorem without success. As proved by riemann in 1851 it states that the integral of any closed differential, such as f(z)dz where f is a continuously differentiable complex function, is zero around any closed path on any branched cover of a portion of the plane, provided that closed path is the complete boundary of a portion of the given surface, and that f is continuously differentiable in that portion of the surface and on the path itself.

i.e. riemann gives the homology version of the theorem already in 1851, although most versions on the web are either without clear hypotheses or restricted to the simply connected case.

i.e. in a simply connected region riemann's hypothesis on the path holds for all closed paths in the region. the key distinction here is that riemann's hypothesis reveals why the theorem is true, as it involves exactly the necessary ingredient for the proof via green's thoerem, and is not only sufficient but apparently also necessary for the theorem to hold.

i could not find a reference for cauchy's original version and hence do not know what hypotheses cauchy used. since in the form known then, it is an immediate corollary of green's theorem, presumably the innovation by these other mathematicians wass merel the application to the case of compelx numbers, which were a new concept in those days.

(the much stronger theorem now usually called cauchy's theorem is apparently due to goursat, who got his degree about 1881, some 20 or 30 years after cauchy's death.)

 

[goldi]

thx mathwonk for urs wonderful explanation.
its greatly appreciated.

How will u prove that a function is continuous at a point z when it is given to be differentiable at that point...
I am confused which definition of continuity to bring in it here...
I welcome yours suggestions and hints.

 

[matt grime]

Any of them will do

if f is differentiable at x then

lim f(x+h)-f(x) = hf'(x) +ho(h)

as h tends to zero the lhs tends to 0 thus f(x+h) tends to f(x) and hence f is continuous at x.

 

[goldi]

Matt but it seems to pretty simple...
we are taking h on the complex plane ,,so h must be complex...Does n't that have any role to play?
and besides on the RHS of the eqn what is this term ho(h)/

 

[hola]

My funnest experience with complex variables: last year, I was bored, and decided to self-study this subject. Anyways, one day in our Calc class, we had a test on trignometric integrals, and instead of messy substitution inside the integrals, I just plugged in the complex representations for sin and cos in and got the answers quickly and sweetly. My teacher was so impressed with me, he gave me a 200/100 on the test! Sweetness...

 

[fourier jr]

favourite part of complex analysis so far... solving real integrals with residues :!)

 

[lurflurf]

My funnest experience with complex variables: last year, I was bored, and decided to self-study this subject. Anyways, one day in our Calc class, we had a test on trignometric integrals, and instead of messy substitution inside the integrals, I just plugged in the complex representations for sin and cos in and got the answers quickly and sweetly. My teacher was so impressed with me, he gave me a 200/100 on the test! Sweetness...

You are a fortunate individual. Many calculus teachers would write "Use of unauthorized method 0!" or "Use of method I do not understand 0!".

 

[Galileo]

I think the reason why the field of complex numbers is so powerful and useful is because complex numbers have many 'geometric properties.' Adding complex numebrs vectors is like adding vectors and multiplying them is like rotating+scaling.

Studying plane geometry with complex numbers instead of vectors is much easier.
Also, you can show that you cannot trisect an angle using a compass and a straightedge using complex numbers.

Other geometric propertie: Reflection principle: $f(\bar z)=\bar{f(z)}$ if f is analytic in a domain which contains the real axis and is symmetric wrt that axis.

Mathematicians in leiden have finished an unfinished print from Escher, showin the Droste effect. They solved it using complex numbers http://escherdroste.math.leidenuniv.nl/index.php?menu=intro

 

[mathwonk]

i had no idea why the cauchy riemann equations should be true until i learned that they simply say the real linear derivative of the function, as a linear map, is also complex linear.

i.e. a 2 by 2 real matrix with entries a,b,c,d, represents the real matrix for complex linear map from C to C, if and only if a = d and b = -c. Then the matrix represents simply multiplication by a - i b, I think.

(Of course I did not tel you where the entries were in the matrix, so I'm probably safe.)

 

[mathwonk]

by the way galileo, is frans oort still there in leiden? chris peters? edouard looienga?

 

[Galileo]

Ehm. Frans Oort is, but he could also be in utrecht. Don't know about the others.

Actually, I don't know any of them, but I couldn't find them on http://www.math.leidenuniv.nl/people/pbase_e.html

 

[mathwonk]

thanks for the link. i found oort and looijenga at utrecht, and van der geer at amsterdam, all old friends.

i did not find chris peters yet. ahh, he's in grenoble. i wonder where murre is? aha! he is emeritus at leiden! thanks again.

somethig is wrong with those links though, those young kids like chris have grey hair in them. of course it has been 20 or 25 years since we met...

oh that reminds me of a fun experience in (several) complex variables, reading the cartan seminars. as i recall, one has the iremann extensions theorem in complex analysis of one variable that a complex function analytic in the punctured disc extends analyticazlly acros the center point iff it is bounded in some nbhd of the puncture.

but in several variables, at least 2, it extends always! i was in leiden speaking on my joint work, showing that the locus of principally polarized abelian varieties of dimension 5, having a singular point on the theta divisor, had exactly two components, and i used this nice result in the writeup.

the nice thing about complex anaoysis is that even if you cannot deal explicitly with what is going on in codimension 2 or more, you can often just forget about it.

 

[Timbuqtu]

Sorry for re-opening this thread. Funny to hear you know Looijenga, because I attended a course on 'complex analysis' last year taught by him. It's a small world...

 

[mathwonk]

to justify reopening the thread here is another comment on the unreasonable usefulness of compelx analysis:

complex numbers form a larger domain in which real numbers are a special case. hence using complex numbers even to study wuestions about real numbers should be expected to yield new insights evena s beoing able to see in two or three dimensions gives an advantage over someone whose vision is, limited to one dimension.

riemann gave a huge boost to the study of prime numbers starting from eulers observation that the product of a certain family of fractions, ranging over all prime numbers, was equal to a certrain sum of otherf ractions ranging over all natural numbers.


since the sums and products were infinite, in fact netiher was convergent. but modifying the exponents of the fractions, by letting them change from 1 to a variable s, allowed convergence for |s| > 1, and then allowed one to use complex analysis to prove the convergent expressions actually had complex differentiable extensions to all numbers except s= 1!


this paper then contained a hypothesis about this function still unsolved to this day, which allows one to use complex path integration and analytic contyinuation to shed light on the distribution of the primes occurring in euler's original product.

looijenga by the way is one of the strongest and most productive algebraic geometers in the world. you were fortunate to study with him.

 

[leright]

take a mathematical physics course or an advanced engineering math course...that will show you the applications.