Search results

  1. M

    Complex derivative and div/curl

    In trying to get an intuition for curl and divergence, Ive understood that in the case of R2, div f(x,y) = 2Re( d/dz f(z,z_)) and curl f(x,y) = 2Im( d/dz f(z,z_)), where f(z,z_) is just f(x,y) expressed in z and z conjugate (z_). Is there any way of proving the fundamental properties of div and...
  2. M

    Using complex description of div and curl in 2d?

    some characters did not write out, f(z,z_) its supposed to say, with z_ being complex conjugate
  3. M

    Using complex description of div and curl in 2d?

    In trying to get an intuition for curl and divergence, Ive understood that in the case of R2, div f(x,y) = 2Re( d/dz f(z,z_)) and curl f(x,y) = 2Im( d/dz f(z,z)), where f(z,z) is just f(x,y) expressed in z and z conjugate (z). Is there any way of proving the fundamental properties of div and...
  4. M

    Tools to analyze sequences

    I need the math tools to understand and analyze sequences and their convergence. I know for example that the fibonacci series can be rewritten such that we can calculate for example nr 153 without knowledge of previous numbers. What math subjects is needed to take care of more complicated...
  5. M

    Question about eigenvectors

    General relation I meant off cause ;\
  6. M

    Question about eigenvectors

    Is there a general between the eigenvectors of a matrix and the row (or column) vectors making up the matrix?
  7. M

    Simplify the proof of different vector calculus identities

    Im talking about all these identities, is there a branch of mathematics that simplifies the proofs of these, and lets me avoid expending the vectors and del operators?
  8. M

    Simplify the proof of different vector calculus identities

    Is there a way to simplify the proof of different vecot calculus identities, such as grad of f*g, which is expandable. And also curl of the curl of a field. Is there a more convenient way to go about proving these relations than to go through the long calculations of actually performing the curl...
  9. M

    Making sense of vector derivatives

    Im trying to understand helmholts decomposition, and in order to do so, I feel the need to understand the different ways to apply the del operator to a vector valued function. The dot product and the cross product between two ordinary vectors are easy to understand, thinking about them as a...
  10. M

    Motivation of sin and cos functions

    Is there a way to motivate the sinus and cosinus functions by looking at their Taylor expansion? Or equivalently, is there a way to see that complex numbers adds their angles when multiplied without knowledge of sin and cos?
  11. M

    Mathematical insight about waves

    Dr Courtney, I understand but your statement implies that we know that e-i2π = 1, and from the definition of eix by its taylor expansion, how can we see on its derivatives that its gonna be periodic and perhaps also how do we see from this definition that it describes a circle? The same question...
  12. M

    Mathematical insight about waves

    I want to have a simple and intuitive explanation of why the sin and cos waves have such a simple repetitive values for their derivatives at a specific point. Their derivative values are also periodic in respect to the derivative order. For example, e^-x is also periodic, but its derivatives are...
  13. M

    Holomorphic functions

    Yes, I know about that condition, but how does that imply that the derivate value in one point is independent of the direction in which it is approached? What does being continuously differentiable have to do with that?
  14. M

    Holomorphic functions

    I was told an analytic complex functions has the same derivation value at z0 (random point) however you approach z0. The cauchy riemann eq. shows that z0 has the same derivate value from 2 directions, perpendicular to each other. However, at least some real functions can have the same derivate...
  15. M

    Sturm Liouville problems

    Thanx, but I meant to integrate the legendre with another function, <f,P(n)>, the inner product. My function is x^2, so the integral/inner product will be <x^2,P(n)>, to find the coefficients for a series-expression of x^2 with legendre as the base.
  16. M

    Sturm Liouville problems

    Hello. I have a big test tomorrow, and there is one thing I can't seem to figure out: In Sturm-Liouville problems, when the legendre polynomials is the solution to the equation, and the boundry-conditions is a function of some sort, I am trying to find the coefficients for expressing the...
  17. M

    Simple integration of bessel functions

    I seek a way to integrate J0, bessel function. I try to use some of the identities I can find, but it takes me no were. Please help!
  18. M

    What is the difference between geometric series and laurent series?

    with ordinary letters: 1/(1-cos(x)) = 1/(1-E(-1)^n*x^(2n)/(2n!)) = 2*E(E((-1)^n*x^(2n-2))/((2n+2)!))^m where both E's (sums) goes from 0 -> inf, and m is for the outer sum.
  19. M

    What is the difference between geometric series and laurent series?

    Here is how: \sum 1/(1-cos(x)) = \sum 1/(1-(\sum(-1)^n(x)^{2n}/(2n!))) = \sum(\sum(-1)^n(x)^{2n-2}/((2n-2)!))^m hope it doesnt look like a mess :)
  20. M

    What is the difference between geometric series and laurent series?

    OK, but the series I come up with doesnt give any z^-1 terms, as it's one taylor expansion of cos, goes from 0 -> inf, and one geometric series of 1/(1-f(x)), goes from 0 -> inf. That is, no -inf. And the cos series only produces even number powered z:s. And is there any easy rules for finding...
  21. M

    What is the difference between geometric series and laurent series?

    Thanks. I tried this way of solving above, and now I get a sum in a sum. And its from 0 -> inf. This isnt a proper laurentseries, is it? Also, I didn't understand the coefficient for powers of z stuff. And choosing number of terms? Can I not just expand it as it is right now, a geometric series...
  22. M

    What is the difference between geometric series and laurent series?

    I don't quite understand a few details here. First, What is the difference between geometric series and laurent series? Than, how do I multiply/divide 2 series with each other? Finally, I have this problem, and I'm really clueless as of what to do. Turn 1/(1-cos(z)) into a laurent series.
  23. M

    The rouché theorem

    I have a problem with applying the rouché theorem to bigger polynomials. Generally, (Az^4 + Bz^3 + Cz^2 + Dz + E) on some annulus k1 < |z-z0| < k2 So, I've tried applying the theorem by the version |f - g| < |f|, and I've chosen g as different terms in the polynomial, starting from z^4...
  24. M

    Need help with complex problems

    YES!! thanx. How come it has to be clockwise?
  25. M

    Need help with complex problems

    Ok, I've tried it now, and I still get it imaginary. sin(az)/(z^2 + 4z + 5) = sin(az)/((z+2-i)(z+2+i)) = (1/2i)*(e^(iaz)/ ((z+2-i)(z+2+i))) - (1/2i)*(e^(-iaz)/ ((z+2-i)(z+2+i))) And now choosing poles, the e^(iaz) needs to have the upper half-plane, and the (-2+i) residue, and the other...
  26. M

    Need help with complex problems

    Ooh, very nice. Trying it out.
  27. M

    Need help with complex problems

    Ok, thanx. How come I can use the exp function in this way, but not the sin function? in other words, why do I need to expand the sin function but not the exp function to get the residue? And also, why does the contour integrals have to be in different half planes?
  28. M

    Need help with complex problems

    Well, so I choose the residue at (-2 + i), and thereby the first and second quadrant. In the (z +2 -i)(z +2 +i) term, the first one gets canceled, and inserting my chosen number in the second gives 1/(2i) (cuz its in the denominator). This is canceled out when I rewrite the sin part. Now I...
  29. M

    Need help with complex problems

    Need help with complex problems!! I have trouble with a residue problem: Integral -inf --> inf sin(ax)/(x^2 + 4x + 5) As both it's poles contains both a real and a complex side, the sin(ax) part gets very ugly when I try to calculate it's residues. And I get a complex answere! Do I just...
Top