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Q: Do I have to consider both problems (real $\xi$ or imaginary $\xi$) totally independently and work hard twice?. A: A single calculation suffices, you could just do the inverse Fourier transform of …

The Riesz transform of the function $f(x,y)$ of two variables reads, $$ \mathcal{R}_x(f(x,y))=\frac{1}{2\pi}\iint \frac{(x-u)f(u,v)}{[(x-u)^2+(y-v)^2]^{3/2}}\,dudv.$$ Let me define $F(x,y)=\partial_x\ …

The Fourier transform $F_\epsilon(x,t)$ has a closed-form expression in terms of hypergeometric functions, $$F_\epsilon(x,t)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{ikx}e^{i\epsilon^4 tk^4}\,dk=$$ $$\q …

McKean's conjecture was proven already in 1967 by Stewart Harris, see Proof that Successive Derivatives of Boltzmann's H Function for a Discrete Velocity Gas Alternate in Sign. See also On the sign of …

Define the vector $X=(y,z)$ and matrices $\sigma_1={{0 1}\choose{1 0}}$, $\sigma_3={{1\; 0}\choose{0\, -1}}$, then the differential equations read $$X_t = -\sigma_3 X_x + \sigma_1 X.$$ No boundary con …

I don't think this statement is correct. As a counter example, take $$g(r)=\frac{e^{-k r} (8 k r-1)}{4 r^{3/2}}$$ which has the desired $r^{-1/2}e^{-kr}$ decay for $r\rightarrow\infty$. It changes sig …

Random walk method for the two‐ and three‐dimensional Laplace, Poisson and Helmholtz's equations (paywall) Random Walk Method for Potential Problems (freely accessible) The random walk method is …

If the time dependence of the potentials is periodic in time, then one enters the field of Floquet wave equations, which is a very active field of study with many real-world and even practical applica …

I agree with @Christian Remling that the product structure of your potential $V(x,t)=g(t)V(x)$ is not helpful in general, but it would help if $g(t)$ is a monotonically decreasing function of time, s …

your second boundary condition is missing a factor $u-u_b$: $$a\frac{\partial u}{\partial n} - \gamma(u-u_b)(\mathbf v \cdot \mathbf n) + \beta(u - u_b) = 0$$ the coefficient $\beta$ gives the stren …