- #1

rhia

- 37

- 0

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?

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In summary, the conversation discusses the differences between real differentiation and integration and their complexities. It is noted that complex differentiation is special because the complex plane is two-dimensional and the limit must be approached from any direction. The function must also be infinitely differentiable in order to be considered analytic. Integration is more sophisticated and should not be thought of as just finding the area. The Riemann-Stieltjes integral is needed for complex line integrals.

- #1

rhia

- 37

- 0

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?

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- #2

matt grime

Science Advisor

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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).

- #3

HallsofIvy

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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").

- #4

mathwonk

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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.

- #5

bombadillo

- 30

- 0

mathwonk said: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 asadding 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.

- #6

abia ubong

- 70

- 0

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

Calculus is a branch of mathematics that deals with the rates of change of different quantities. It is divided into two main branches: differential calculus, which studies the rate of change of a function, and integral calculus, which studies the accumulation of quantities over a given interval.

Calculus can be considered difficult because it requires a strong foundation in algebra and trigonometry, as well as the ability to think abstractly and visualize complex concepts. It also involves a lot of problem-solving and critical thinking skills.

Differentiation is the process of finding the derivative of a function, which represents the rate of change of that function. Integration, on the other hand, is the process of finding the antiderivative of a function, which represents the accumulation of that function over a given interval.

Understanding the fundamentals of calculus is important because it is the basis for many other fields of mathematics, as well as physics, engineering, and other sciences. It also helps in developing critical thinking and problem-solving skills.

Improving your understanding of complex calculus requires practice and patience. Start by mastering the basic concepts and techniques, and then move on to more challenging problems. It is also helpful to seek out additional resources, such as textbooks, online tutorials, and practice problems to reinforce your knowledge.

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