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MuPAD^{®} provides many special functions commonly used in
engineering and science. MuPAD uses standard mathematical notations
for special functions. If you do not recognize a notation, see Mathematical Notations
Uses in Typeset Mode.

Particular parameter choices can simplify special functions. Often MuPAD handles such simplifications automatically. For example, the following parameters reduce hypergeometric functions to elementary functions:

hypergeom([], [], z); hypergeom([1], [], z); hypergeom([a], [], z)

MuPAD does not automatically simplify some functions. For example, it does not automatically simplify the Meijer G special function:

meijerG([[], []], [[1], []], z)

The general simplification functions, `simplify`

and `Simplify`

, represent
this expression in terms of elementary functions:

simplify(meijerG([[], []], [[1], []], z))

MuPAD also does not use automatic simplifications for many expressions involving special functions. Suppose you get an expression containing the Fresnel sine integral function:

2*fresnelS(z) + fresnelS(-z)

To apply the reflection rule `fresnelS(-z) = -fresnelS(z)`

and
simplify this expression, explicitly call one of the general simplifiers:

simplify(2*fresnelS(z) + fresnelS(-z))

Particular values of parameters can reduce more general special
functions to expressions containing simpler special functions. For
example, reduce `meijerG`

to the hypergeometric functions:

Simplify(meijerG([[1/3, 1/3, 3/2], []], [[0], [-2/3, 4/3]], z))

The following choice of parameters expresses `meijerG`

in terms of
the Bessel functions:

simplify(meijerG([[], []], [[1], [1]], z))

MuPAD supports expansions of expressions containing special
functions. The resulting expressions can involve the original or additional
special functions, or both. For example, the `expand`

command expresses the beta function
by gamma functions:

beta(x + 1, y) = expand(beta(x + 1, y))

When you expand the gamma function, MuPAD expresses it in terms of gamma functions:

gamma(5*x + 1) = expand(gamma(5*x + 1))

When solving equations (especially ordinary differential equations), you often get the results in terms of special functions. For example, consider the following differential equation:

eq := diff(y(x), x, x) - x^3*y(x) = 0

For this ODE, the solver returns the result in terms of the Bessel functions:

S := solve(ode(eq, y(x)))

To verify correctness of the returned solution, try substituting
it into the original equation by using `evalAt`

or its shortcut `|`

. You
get the following long and complicated result that still contains
the Bessel special functions. MuPAD does not automatically simplify
this result:

eq | y(x) = S[1]

Simplifying this expression proves the correctness of the solution:

simplify(eq | y(x) = S[1])

The `testeq`

command
serves best for verifying correctness of the solutions. The command
automatically simplifies expressions on both sides of the equation:

testeq(eq | y(x) = S[1])

For more information see Testing Results.

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