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  "body": "\n## Definition\n\nLet $\\mathbb{H}$ be the [[quaternions|0GE4DVQKR8]], the real [[division ring|0EA5BTA0CX]] with basis $1, i, j, k$ and $i^2 = j^2 = k^2 = ijk = -1$.\nWrite $q = x_0 + x_1 i + x_2 j + x_3 k$ with $x_0, \\ldots, x_3 \\in \\R$, and identify $\\mathbb{H}$ with $\\R^4$.\nLet $U \\subset \\mathbb{H}$ be open.\nA function $f \\colon U \\to \\mathbb{H}$ with continuous first [[partial derivatives|09EGC6F8PT]] is *left regular*, or *Fueter regular*, if\n$$\\frac{\\partial f}{\\partial x_0} + i \\frac{\\partial f}{\\partial x_1} + j \\frac{\\partial f}{\\partial x_2} + k \\frac{\\partial f}{\\partial x_3} = 0 \\quad \\text{on } U.$$\n*Quaternionic analysis* is the study of functions $U \\to \\mathbb{H}$ by analogy with complex analysis, and above all of regular functions.\n\n## Remarks\n\nExample: $f(q) = x_1 - i x_0$ is left regular, since $\\partial_{x_0} f + i \\partial_{x_1} f = -i + i = 0$ and the other two derivatives vanish.\n\nNon-example: $f(q) = q$ is not left regular.\nThe sum is $1 + i \\cdot i + j \\cdot j + k \\cdot k = 1 - 3 = -2 \\ne 0$.\nSo, unlike in complex analysis, the identity function and polynomials in $q$ are in general not regular.\n\nWrite $\\bar\\partial = \\partial_{x_0} + i \\partial_{x_1} + j \\partial_{x_2} + k \\partial_{x_3}$ and $\\partial = \\partial_{x_0} - i \\partial_{x_1} - j \\partial_{x_2} - k \\partial_{x_3}$.\nThen $\\partial \\bar\\partial$ is the [[Laplacian|0E0B6RA8K0]] of $\\R^4$, so each component of a regular function of class $C^2$ is harmonic.\nRight regular functions are defined with the units multiplied on the right of the derivatives.\n",
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