xy = cos (x + y) AND OTHER EQUATIONS
THAT ARE SURPRISINGLY EASY TO PLOT
BY MICHAEL JEWESS
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This
work was published in Mathematical Gazette, 2024, 108, 1-11, open access doi https://doi.org/10.1017/mag.2024.2.
Abstract
The
equation xy = cos (x + y)
at first sight looks as if it would be exceptionally tedious to plot without a
computer, and indeed was specifically once stated to be “impossible to solve ... in any
effective sense”. It is indeed not
soluble in any closed form but it is parametrisable so that it could have be
plotted without the use of a computer* when the latter were generally
unavailable (before ca 1950) or inconvenient to use (up to the mid-1970s). This is a member of a substantial class of equations
that can be plotted likewise:
f(x, y) = S(g(x, y)),
where
(i) S does not include any non-elementary
functions; and
(ii) f and g are such that the simultaneous
equations f(x, y) = P2 and g(x, y) = P1 can be
solved for x and y in terms of P1
and P2 without use of
non-elementary functions, iterative numerical methods, or graphical methods.
* eg using a hand-cranked four-function calculator and
mathematical tables
CONTACT
Use this link symmetry to contact Dr Michael
Jewess.