Download 3.4.4 Equations Involving Functions

3.4 Manipulating Equations
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3.4.4 Equations Involving Functions
Solve is primarily intended for solving algebraic equations. However, at least with the default option setting InverseFunctions -> True, Solve will attempt to solve some equations that involve other functions.
This solves a simple transcendental
equation for x.
In 1]:= Solve ArcSin x] == a, x]
Out 1]= {{x -> Sina]}}
In practice, very few equations involving transcendental functions can be solved exactly in symbolic
form. And even in those cases where some solutions can be found, it is often impossible to get all the solutions.
Mathematica prints a warning indicating
that solutions are lost in this case.
In 2]:= Solve Sin x] == a, x]
Solve::ifun:
Warning: inverse functions are being used by Solve, so some
solutions may not be found.
Out 2]= {{x -> ArcSina]}}
An equation like sin(x) = a in principle has an infinite number of solutions for x differing by multiples
of 2 . However, even with the setting InverseFunctions -> True, Solve gives you only one of the
possible solutions.
While all the solutions to this particular equation would be easy to parameterize, most such equations
yield much more complicated sets of solutions. With simultaneous trigonometric equations, for example,
it is common to end up needing solutions to arbitrary Diophantine equations, which cannot in general be
found by any finite procedure.
The fact that Mathematica generates only some of the solutions to each equation means that if you try to
solve several simultaneous equations, no solutions may be found, even though solutions do in fact exist.
With an unknown function f,
Mathematica gives a formal solution in
terms of the inverse function of f.
In 3]:= Solve f x] == a, x]
Solve::ifun:
Warning: inverse functions are being used by Solve, so some
solutions may not be found.
Out 3]= {{x -> f
This shows the structure of the inverse
function.
(-1)
a]}}
In 4]:= InputForm %]
Out 4]//InputForm= {{x -> InverseFunctionf, 1, 1]a]}}
If you ask Mathematica to solve an equation involving an unknown function, it will try to construct a
formal solution to the equation in terms of inverse functions. In doing this, Mathematica effectively assumes that the function indeed has a unique inverse. If there are several possible inverses, as in the case
of a function like Sin, Mathematica will inevitably lose solutions in such a case.
Web sample page from The Mathematica Book, Second Edition, by Stephen Wolfram, published by Addison-Wesley Publishing Company (hardcover ISBN 0-201-51502-4; softcover ISBN 0-201-51507-5). To order Mathematica or this book contact Wolfram Research: [email protected];
http://www.wolfram.com/; 1-800-441-6284.
 1991 Wolfram Research, Inc.
Permission is hereby granted for web users to make one paper copy of this page for their personal use. Further reproduction, or any copying of machine-readable files (including this one) to any server computer, is strictly prohibited.
3. Advanced Mathematics in Mathematica
700
InverseFunctionf]
InverseFunctionf, n]
InverseFunctionf, n, tot]
the inverse function of f
the inverse function of f with respect to its nth argument
the inverse function of f when the total number of arguments
is tot
Inverse functions.
Here is the inverse function of f
evaluated with argument a.
In 5]:= InverseFunction f] a]
Applying f to the previous expression
gives a.
In 6]:= f %]
(-1)
Out 5]= f
a]
Out 6]= a
Web sample page from The Mathematica Book, Second Edition, by Stephen Wolfram, published by Addison-Wesley Publishing Company (hardcover ISBN 0-201-51502-4; softcover ISBN 0-201-51507-5). To order Mathematica or this book contact Wolfram Research: [email protected];
http://www.wolfram.com/; 1-800-441-6284.
 1991 Wolfram Research, Inc.
Permission is hereby granted for web users to make one paper copy of this page for their personal use. Further reproduction, or any copying of machine-readable files (including this one) to any server computer, is strictly prohibited.