Solution
= Solution
At zero, the <Fourier transform> is $\hat f(0)=\int f=1$, so $\lambda f(0)=1$. In particular $\lambda\ne0$ and $f(0)=1/\lambda$. The <Fourier inversion theorem> at zero also gives
$$
f(0)=\frac1{2\pi}\int_{\mathbb R}\hat f(s)\,ds=\frac\lambda{2\pi}\int_{\mathbb R}f(s)\,ds=\frac\lambda{2\pi}.
$$
Equating the two values yields $\boxed{\lambda^2=2\pi}$. Since the explicit normalized <Gaussian function> in part (i) has $f(0)>0$, here $\lambda=\sqrt{2\pi}$.