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  "body": "\n## Example\n\nFix a parameter $\\sigma \u003e 0$ and put $\\kappa_\\sigma = e^{-\\sigma^2/2}$.\nThe *Morlet wavelet* is the complex-valued function\n$$\\psi_\\sigma(t) = c_\\sigma \\, \\pi^{-1/4} e^{-t^2/2} \\left(e^{i\\sigma t} - \\kappa_\\sigma\\right), \\qquad t \\in \\R,$$\nwhere $c_\\sigma \u003e 0$ is the constant that makes $\\norm{\\psi_\\sigma}_2 = 1$.\nIt is a plane wave $e^{i\\sigma t}$ under a Gaussian envelope, with a small correction term.\n\nCheck of the zero mean.\nSince $\\int_{\\R} e^{-t^2/2} e^{i\\sigma t} \\, dt = \\sqrt{2\\pi} \\, e^{-\\sigma^2/2}$ and $\\int_{\\R} e^{-t^2/2} \\, dt = \\sqrt{2\\pi}$, the integral of $\\psi_\\sigma$ is $c_\\sigma \\pi^{-1/4} \\sqrt{2\\pi} \\left(e^{-\\sigma^2/2} - \\kappa_\\sigma\\right) = 0$.\nIts Fourier transform is a combination of two Gaussians and vanishes at frequency $0$.\nTogether with the Gaussian decay, this gives the [[admissibility condition|2BBB8NY5ZA]], so $\\psi_\\sigma$ can be used in the [[continuous wavelet transform|57FR21NDGV]].\n\n## Remarks\n\nThe correction $\\kappa_\\sigma$ is tiny when $\\sigma$ is large; for $\\sigma \\ge 5$ it is below $10^{-5}$, and in practice it is often dropped.\nWithout the correction, the function $e^{-t^2/2} e^{i\\sigma t}$ has nonzero integral, so it is not admissible in the strict sense.\nThe Morlet wavelet does not generate an orthonormal basis; it is used for the continuous transform and for time–frequency analysis of signals.\nIts real part is even and its imaginary part is odd.\nSince the wavelet is complex-valued, the modulus of the transform gives an amplitude and the argument gives a phase.\n\n## Explanations\n\n### L1 human\n\nThe Morlet wavelet is a short burst of oscillation: a wave of frequency $\\sigma$ that fades out like a bell curve on both sides.\nComparing a signal with stretched and shifted copies of it tells how much of each frequency is present near each time.\nThe small constant subtracted from the wave makes the average of the function exactly zero, which a wavelet needs.\n",
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