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Let $\theta$ be such that $0<\theta<1/2$. Show that $$F(t,x)=\int_{-\infty}^{+\infty}\exp[i\tau t-(i\tau)^{1/2}x-(i\tau)^\theta]d\tau$$ defines a $C^\infty$ function in $\mathbb{R}^2$ which solves the homogeneous heat equation $F_t=F_{xx}$. Also show that $F$ can be written as $$\int_{-a-i\infty}^{a+i\infty}\exp[-zt-z^{1/2}x-z^\theta]dz$$ where complex integration is performed over the vertical straight line $\text{Re} z=-a$ where $a>0$. How to proceed with this question ?

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