How do complex differentiation and integration differ from real calculus?

[rhia]

How is it different from the real differentiation and integration?
There are so many details that I am finding it hard to understand.
Is there a better way to understand especially Integration of Complex functions?

 

[matt grime]

Well, the domain is different for a start.

The reason why complex differentiation is special is this:

C is both a "1 parameter space" or it is a 1-d space, whatever, susing C as the groudn field, and it is a 2-d real space.

C = RxR, the set of ordered pairs of real numbers with z = x+iy identified with (x,y)

so when we do lim h tends to 0 of [f(z+h) - f(z)]/h, we can also think in terms of what we want to happen thinking of

f(z) = f(x,y) = u(x,y)+iv(x,y)


there is a whole thread on this in this very subforum. try searching for it.

anyway, it turns out the proper definition for complex differentiation is one where, treating u and v as real valued functions from RxR, we have the cauchy riemann equations satisfied. goolgle for these (include the word wolfram, as ever).

 

[HallsofIvy]

One important difference is that the complex plane is two dimensional. In order that a function be differentiable, it must be true that $$lim_{x->0} \frac{f(x+h)- f(x)}{h}$$ exists. In functions of a real variable, that only means that the two limits "from above" and "from below" must exist. In functions of a complex variable, that means that the limit as you approach from any direction, any line, any curve, must give the same result.
A result of that is that if a function of a complex variable has a continuous derivative it must be infinitely differentiable (actually even more- "analytic").

 

[mathwonk]

differentiation is easy to explain. A complex valued function ,of a complex variable is equivalent to a function from R^2 to R^2. A derivative is alinear approximation.

If the function is continuiously differentiable in the real sense, then the function is also complex differentiable if and only if the real linear approximation is actually complex linear as well.

integration is alittle more sophisticated. Try to get over thinking of integration as area, and just as adding up something.

 

[bombadillo]

differentiation is easy to explain. A complex valued function ,of a complex variable is equivalent to a function from R^2 to R^2. A derivative is alinear approximation.

If the function is continuiously differentiable in the real sense, then the function is also complex differentiable if and only if the real linear approximation is actually complex linear as well.

integration is alittle more sophisticated. Try to get over thinking of integration as area, and just as adding up something.

It's the Riemann-Stieltjes integral that's needed for complex line integrals (just as it's needed for the ordinary line integrals of vector analysis). On a contour, at a given point z*, we're multiplying the complex number f(z*) by the complex number z**-z* (where z** is a point close to z*: z** = z*+delta z); we sum over all these products over the contour, and then look at the limit as z**-z* tends uniformly to zero, for all z*,z**.

I've seen many introductory books gloss over these small technical problems, and treat the complex integral as an exact analogue of our chum the Riemann integral.

 

[abia ubong]

hey mathwonk and hallsofivy u are both good, send a private 2 me u both