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Let us define $\epsilon_0=0.1$, so that $ I(x)=- x^{-1}\Im\int_{\mathbb R}\frac{H(t-1-\epsilon_0)e^{itx}}{t^{2-\epsilon_0}\ln t} dt. $ We get for $x>0$, $$ I(x)=-x^{-1}\int_{x(1+\epsilon_0)}^{+\infty} …

$\phi=\Re \phi+i\Im \phi$. You have assumed $\Re \phi\ge 0$ and you deal with a complex phase function. Note that the standard notation is not yours, since what is usually called the real stationary p …

$$ I(\tau)=\int_{\mathbb R}e^{-\pi\frac{\tau}{\pi} x^2}e^{-2i\pi \tau x (1-\frac{1}{2i\pi})}dx= (\frac{\pi}{\tau})^{1/2}e^{-\pi\frac{\pi}{\tau} \tau^2 (1-\frac{1}{2i\pi})^2}= (\frac{\pi}{\tau})^{1/2}e …

With $J_n$ standing for the Bessel function of first kind, $n\in \mathbb N$, I define $$ f_n (\rho) =\int_0^π J_n(\rho \sin \theta) \sin \theta \ d\theta.$$ Assuming $1\ll\rho\ll n$, I would like to f …

Let $J_\nu$ be the standard Bessel function of the first kind and let $x_\nu$ be its smallest zero. Is there a simple reference or result for the asymptotic expansion of $x_\nu$ when $\nu$ goes to $+\ …

You have $ I(\lambda, x)=x\cdot\int_{\mathbb S^{n-1}} ye^{i \lambda x\cdot y} d\sigma(y)=x\cdot J(x,\lambda) $ and you claim that for $\vert x\vert \lambda \ge 1$, you have $$ J(x,\lambda)=O((\vert …

Let me assume that $u$ is depending only on the $y$ variable, that $f$ is smooth and depends also only on $y$ and is such that $$ \Im f\ge 0,\quad f(0)=0, df(0)=0,\quad \det f''(0)\not=0. $$ Then the …

. $$ Both asymptotics are obtained from Stirling's formula. …