1 \documentclass[11pt]{article}
8 The Landau function $L$ is a function of $x$ with the parameters
9 $\Delta_p$ and $\xi$. It can be written as
11 \phi(u) &= \frac1\pi\int_0^\infty dt\,\sin(\pi t) e^{-t\log{t}-ut}
13 L(x;\Delta_p,\xi) &= \frac1\xi
14 \phi\left(\frac{x-\Delta_p}{\xi}\right)\label{eq:land}
17 A Landau convolved with a Gaussian is given by
19 L_{G}(x;\Delta_p,\xi,\sigma) &= \int_{-\infty}^{+\infty}
20 ds\, L(s;\Delta_p,\xi) \frac{1}{\sqrt{2\pi}\sigma}
21 e^{-\frac{(x-s)^2}{2\sigma^2}}\nonumber\\
23 \frac{1}{\sqrt{2\pi}\sigma\pi\xi}\int_{-\infty}^{+\infty}ds\,\int_0^\infty
24 dt\,\sin(\pi t)e^{-t\log{t}-\frac{s-\Delta_p}{\xi}t}
25 e^{-\frac{(x-s)^2}{2\sigma^2}}\nonumber\\
26 &= \frac{1}{\sqrt{2\pi}\sigma\pi\xi}\int_0^\infty
27 dt\,\sin(\pi t) e^{-t\log{t}+\frac{\Delta_p}{\xi}t}\int_{-\infty}^{+\infty}
28 ds\,e^{-\frac{(x-s)^2}{2\sigma^2}}e^{-\frac{st}{\xi}}\label{eq:lg}
31 Carrying out the inner integral of \eqref{eq:lg},
34 \int_{-\infty}^{+\infty}
35 ds\,e^{-\frac{(x-s)^2}{2\sigma^2}}e^{-\frac{st}{\xi}} &=
36 \int_{-\infty}^{+\infty}
37 ds\,e^{\frac{-s^2-x^2+2xs}{2\sigma^s}-\frac{st}{\xi}}\nonumber\\
38 &= \int_{-\infty}^{+\infty}
39 ds\,e^{-\frac{s^2}{2\sigma^2}+\left(\frac{2x}{2\sigma^2}-\frac{t}{\xi}\right)s-\frac{x^2}{2\sigma^2}}\nonumber\\
40 &= \int_{-\infty}^{+\infty}
41 ds\,e^{-\frac{1}{2\sigma^2}\left[s^2+\left(\frac{2\sigma^2t}{\xi}-2x\right)s\right]-\frac{x^2}{2\sigma^2}}\nonumber\\
42 \intertext{Completing the square}
44 &= \int_{-\infty}^{+\infty}
45 ds\,e^{-\frac{1}{2\sigma^2}\left[s^2+2\left(\frac{\sigma^2t}{\xi}-x\right)s
46 +\left(\frac{\sigma^2t}{\xi}-x\right)^2
47 -\left(\frac{\sigma^2t}{\xi}-x\right)^2\right]-\frac{x^2}{2\sigma^2}}\nonumber\\
48 &= \int_{-\infty}^{+\infty}
49 ds\,e^{-\frac{1}{2\sigma^2}\left[s + \left(\frac{\sigma^2t}{\xi}-x\right)\right]^2
50 +\frac{1}{2\sigma^2}\left(\frac{\sigma^2t}{\xi}-x\right)^2
51 -\frac{x^2}{2\sigma^2}}\nonumber\\
52 \intertext{Recognizing this as a Gaussian integral, we get}
54 %% integrate(%e^(-1/(2*sigma^2)*(s+sigma^2*t/xi-x)^2 +
55 %% 1/(2*sigma^2)*(sigma^2*t/xi-x)^2-x^2/(2*sigma^2)),s,-inf,inf);
57 e^{-\frac{1}{2\xi^2}\left(2tx\xi-\sigma^2t^2\right)}\nonumber\\
58 &= \sqrt{2\pi}\sigma e^{\frac{t}{2\xi}\left(\frac{\sigma^2t}{\xi}-2x\right)}
62 Substituting \eqref{eq:inner} back into \eqref{eq:lg} yields
65 L_{G}(x;\Delta_p,\xi,\sigma) &= \frac{1}{\sqrt{2\pi}\sigma\pi\xi}
66 \int_0^{\infty}dt\,e^{-t\log{t} + \frac{\Delta_p t}{\xi}}\sin{\pi t}
67 \sqrt{2\pi}\sigma e^{\frac{t}{2\xi}\left(\frac{\sigma^2t}{\xi}-2x\right)}\nonumber\\
68 &= \frac{1}{\pi\xi}\int_o^{\infty}dt\,e^{-t\log{t} + \frac{\Delta_pt}{\xi}+
69 \frac{t}{2\xi}\left(\frac{\sigma^2t}{\xi}-2x\right)}\sin{\pi
71 &= \frac{1}{\pi\xi}\int_o^{\infty}dt\,e^{-t\log{t} - \frac{(x
72 -\Delta_p)t}{\xi} + \frac12\left(\frac{\sigma}{\xi}t\right)^2}\sin{\pi
76 Defining $v=\frac{\sigma}{\xi}$, and
79 \Phi(u;v) &= \frac{1}{\pi}\int_0^{\infty}dt\,e^{-t\log{t} - ut +
80 \frac12v^2t^2}\sin{\pi t}\label{eq:Phi}\quad,
82 we can write \eqref{eq:lg} as
84 L_{G}(x;\Delta_p,\xi,\sigma) &=
85 \frac1\xi\Phi\left(\frac{x-\Delta_p}{\xi},\frac\sigma\xi\right)\label{eq:lg2}
87 The normalization follows from the fact that
88 $u=\frac{x-\Delta_p}{\xi}$ and $\frac{dv}{du}=f(u)$ so that
90 du &= \frac{dx}{\xi}\quad\text{and}\\
91 \frac{dv}{dx/\xi} &= f\left(\frac{x-\Delta_p}{\xi}\right)\\
93 \frac{dv}{dx} &= \frac1\xi f\left(\frac{x-\Delta_p}{\xi}\right)
98 %% diff(integrate(1/%pi*%e^(-t * log(t)-u*t+1/2*v^2*t^2)*sin(%pi *
102 Differentiating \eqref{eq:Phi} with respect to $u$ gives
106 \int_0^{\infty}dt\,e^{-t\log{t}-tu+\frac12t^2v^2}t\sin{\pi t}
109 Now turning to the maximum of \eqref{eq:phi} which is known to be at
110 $u=u_p\neq0$ we get that the most probable value $x=x_p$ of
111 \eqref{eq:land} is given by
113 \frac{x_p-\Delta_p}{\xi} &= u_p\nonumber\\
115 x_p &= u_p\xi + \Delta_p\quad\text{and}\nonumber\\
116 \Delta_p &= x_p - \xi u_p
119 The maximum of \eqref{eq:Phi} must be a function solely of $v$, and we
120 have the most probable value $x=x'_p$ of \eqref{eq:lg2}
122 f(v) &= \frac{x'_p-\Delta_p}{\xi} \nonumber\\
123 &= \frac{x'_p - x_p + \xi u_p}{\xi} \nonumber\\
124 f(v) - u_p &= \frac{x'_p - x_p}{\xi}\nonumber\\
125 x_p'-x_p &= \xi\left(f(v)-u_p\right)\nonumber\\
126 x_p' &= \xi\left(f(v)-u_p\right)+x_p
127 \intertext{and it follows that}
135 % LocalWords: convolved