In mathematics, the Gibbs phenomenon (also known as ringing artifacts, named after the American physicist J. Willard Gibbs) is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function f behaves at a jump discontinuity: the nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as the frequency increases, but approaches a finite limit.
In MRI, artifactual ripples parallel to abrupt and intense changes are caused by the Fourier transformation.

See Gibbs Artifact, Truncation Artifact, Ringing Artifact Reduction.

An imaginary number is a component of a complex number (satisfying i2 = - 1). In MRI, complex data are used for example in the Fourier transforms.

An integral is a mathematical object that can be interpreted as an area or a generalization of area. A number computed by a limiting process in which the domain of a function, often an interval or planar region, is divided into arbitrarily small units, the value of the function at a point in each unit is multiplied by the linear or areal measurement of that unit, and all such products are summed (summation in the limit). In MRI for example this mathematical function is used in the Fourier transformation.

Usual shape of the lines in a NMR spectrum, characterized by a central peak with long tails; proportional to 1/[(1/T2)2 + (f - fo)2], where f is frequency and fo is the frequency of the peak (i.e., central resonance frequency). A Lorentzian function is the Fourier transformation of a decaying exponential.