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Lapsed engineer/scientist (now a patent practitioner) seeking to expand his math skills. Most of those skills lay in areas useful in optics such as sums & integrals (like the one below), complex analysis, differential equations, transforms, and data analysis.

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"Wir müssen wissen, wir werden wissen." - David Hilbert

The picture on the left represents the integration region for the following integral of functions of finite support:

$$T(\mathbf{u}',\mathbf{u}'') = \int_{\mathbb{R}^2} d^2 \mathbf{\sigma} \; S(\mathbf{\sigma}) P(\mathbf{u}'+\mathbf{\sigma}) P^*(\mathbf{u}''+\mathbf{\sigma})$$


1d
comment Mathematics needed in the study of Quantum Physics
@Victor: My pleasure! I assure you as someone who worked from the original notes and has worked every problem in that book, it will give you an incredible preparation. It is no coincidence that the math that I am most expert in is this sort of math; Prof. Holland was an incredible teacher.
1d
revised Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$
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revised Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$
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revised Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$
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answered Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$
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revised Mathematics needed in the study of Quantum Physics
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answered Mathematics needed in the study of Quantum Physics
1d
answered Minimize total area of a square and triangle made of 13m long wire
1d
comment Bessel Functions Integral Representation proof
@Incognito: and you'd be wrong. Look, the exercise requires some careful handling of imaginary quantities. Work through the steps I provided in detail, especially the conversion to the cosine plus phase. Use the exponential definitions of the cosine and sine to understand the imaginary arguments.
1d
comment Bessel Functions Integral Representation proof
$$\arctan{i y} = i \operatorname{arctanh}{y}$$ If you see how the vertical portions cancel, then as the contour is a rectangle, it does reason that the integrals along the horizontal portions also must cancel, and therefore...
1d
comment Solving integral $ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x $
Perhaps try $$\frac1{1+x+\sqrt{1+x+x^2}} = \frac{1+x-\sqrt{1+x+x^2}}{x}$$
1d
comment Bessel Functions Integral Representation proof
@Incognito: It's the complex version of a standard trick used to express a weighted sum of cosine and sine in in terms of a cosine of something plus a phase. The phase is imaginary and turns out to be the tanh term. You can neglect the imaginary phase for exactly the reason I posted, because of Cauchy's theorem on the contour integral I wrote. BTW contour integration is merely used to prove we can neglect the imaginary phase; it is not used so much in evaluating the actual integral. For that, I rely on the well-known representation of $J_0$.
1d
answered Bessel Functions Integral Representation proof
1d
comment Solve the following wave equation
Consider $u_s(x,t) = t + x t$. Note that this satisfies the wave equation (as all functions linear in $x$ and $t$ must) and reproduce the boundary condition above. Then it's no great shakes to subtract this function from a general solution to get a new solution $v$ that has zero BC's. To get the original solution $u$, just add back $u_s$.
2d
answered Close fourier transforms implies close time domain functions?
2d
comment What is the integral $\int x^t/\Gamma(1+t) \, dt$? (In general: relation between series and integrals)
@GottfriedHelms: try to deduce the behavior of $a_k k!$ for large $k$; that should be interesting.
2d
answered What is the integral $\int x^t/\Gamma(1+t) \, dt$? (In general: relation between series and integrals)
2d
comment What is the integral $\int x^t/\Gamma(1+t) \, dt$? (In general: relation between series and integrals)
Your $g_1(x)$ should equal $$\frac1{\log{\frac1{x}}}$$
2d
comment Approximate an integral
You're not really approximating an integral, but an integrand.
2d
comment Calculate the following Integral (Please Help)
Expand the log into a Taylor series about $x-x^2$, perform the integration, and then recognize the resulting series as an expansion of a (sort of) well-known function evaluated at a particular value.