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...
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...
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...
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...
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...
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?
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...
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...
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...
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.
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...
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...