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  "body": "\n## Summary\n\nQuaternionic analysis studies functions of a quaternion variable with quaternion values, by analogy with complex analysis.\nThe naive analogue of the complex derivative is too restrictive.\nSo the central class is that of Fueter regular functions, the solutions of a quaternionic Cauchy–Riemann equation.\nThese functions have an integral formula like Cauchy's and are harmonic in $\\R^4$.\nA wider theory, [[Clifford analysis|DKVSAZQWF7]], extends these ideas to all dimensions.\n\n## Explanations\n\n### L1 human\n\nThe quaternions extend the complex numbers with three imaginary units $i, j, k$, but their multiplication is not commutative.\nIn complex analysis, a function is holomorphic when it has a derivative $\\lim_{h \\to 0} (f(z + h) - f(z))/h$.\nFor quaternions this limit exists only for affine functions, so this road leads nowhere.\nInstead one copies the Cauchy–Riemann equations: one adds the four partial derivatives, each multiplied by $1, i, j$ or $k$, and asks the sum to be $0$.\nSurprisingly, even $f(q) = q$ fails this test, while some functions of the real coordinates pass it.\n",
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