Analytic functions and integration

In summary, the conversation discusses the concept of analytic functions and their integrals over different paths. It is stated that if a function is analytic in a simply connected region, then the integral from one point to another is independent of the path joining the two points in the domain. However, for a function that is not analytic, the integrals over different paths may not be the same. Additionally, it is mentioned that the orientation of the normal vector does not matter for scalar functions, but it does for vector fields. The concept of Jacobian determinant and Gramian determinant is also briefly mentioned.
  • #1
physicsjock
89
0
I have an example I want to clearify,

Let C be a semicircle from 2 to -2 which passes through 2i and let T be a semicircle from 2 to -2 which passes through -2i.

If you take the integral of z^2 dz around both paths the is the same, as the function is analytic so the integral is independant of the path,

However if you take the integral of the conjugate of z over these to paths you get

[PLAIN]http://img6.imageshack.us/img6/7467/unledpcv.jpg [Broken]
 
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  • #2
[tex] e^{-it} e^{it} = e^0 = 1 [/tex]
 
  • #3
Oh okay,

So they're both i4pi, but like, what does it mean that the integrals are the same over different paths and the function is not analytic?
 
  • #4
You're integrating the same constant from 0..pi and 0..-pi, so they are negative to each other and will cancel each other out.
 
  • #5
"If f(z) is analytic in a simply connected region then the int from one point to another is independent of the path joining the two points in the domain."

So here f(z) isn't analytic, is the only reason the integrals are the same the trivial one?

Because zbar isn't analytic at all
 
  • #6
No dude, I'm trying to tell you that you made a mistake in your calculation, so the integrals are NOT the same. Check your calculation please! The first integral should give 4 pi i and the second one should give -4 pi i
 
  • #7
Oh right, phew,

Thanks man

hey could i ask a small question,

For scalar functions,

is the orientation of the normal vector relavent? the tu X tv,
where h(u,v) is the parametrization, tu tv its partial derivatives
because when you take the surface integral you take f(h(u,v))||tu X tv||dudv, so the orientation really doesn't matter right?

like for vector fields its the dot product of the two so the orientation of the vector does matter
 
  • #8
Yes, that's correct, the orientation of tu x tv doesn't matter if the function is scalar. What matters, of course, is the relative orientation of tu and tv, but that's pretty obvious. You know that |tu x tv| du dv is the Jacobian determinant in case you have different coordinates (like cylindrical or spherical), right? In case you're dealing with a manifold, you have the Gramian determinant.
 
  • #9
Sweet thanks a lot Leon
 

What are analytic functions?

Analytic functions are complex-valued functions that can be represented by a convergent power series. They are defined as functions that have a derivative at every point in their domain.

What is integration?

Integration is a mathematical process of determining the area under a curve or finding the accumulation of a quantity over an interval. It is the reverse process of differentiation and is commonly used in calculus and other areas of mathematics.

How are analytic functions and integration related?

Analytic functions are important in integration because they have a well-defined derivative and can be easily integrated. In fact, all analytic functions are integrable, which means they can be integrated using various techniques such as the Fundamental Theorem of Calculus.

What are some common techniques for integrating analytic functions?

Some common techniques for integrating analytic functions include substitution, integration by parts, and partial fraction decomposition. These techniques are used to simplify the integrand and make it easier to evaluate the integral.

Can analytic functions be integrated using numerical methods?

Yes, analytic functions can be integrated using numerical methods such as the trapezoidal rule, Simpson's rule, and Gaussian quadrature. These methods are used when the integrand cannot be integrated analytically or when the integral is too complex to solve by hand.

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