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Each of the elementary functions of Liouville and Ritt is infinitely differentiable within the elementary functions. That is because the elementary functions form a differential field that is closed under differentiation.

The following are two examples of elementary functions with an n-th derivative which is predictable.

Let $m$, $n$ $\in\mathbb{N}_{+}$.

Your elementary functions shall be infinitely often integrable within the elementary functions. That means, all $n$-th degree antiderivatives of this functions have to be infinitely often differentiable within the elementary functions.

a)

A first simple class of such elementary functions are the polynomials. The antiderivative of a polynomial is again a polynomial. Therefore each $n$-th degree antiderivative of a polynomial is a polynomial again.

b)

A second simple class of such elementary functions, $f$, are the elementary functions having one $n$-th degree derivative that is $f$ again:

$$\frac{d^{n}f(x)}{dx^{n}}=f(x)$$

Because the $n$-th degree antiderivative is an elementary function, all $m$-th degree antiderivatives with $m<n$ are elementary functions again. These differential equations can be easily solved.

$ $

Also the Liouvillain functions form a differential field that is closed under differentiation. Therefore they are also infinetly differentiable in the Liouvillian functions. The elementary functions are a subfield of the Liouvillian functions.

Each function from a set which can be generated by successive integration from a self-differentiable function, e.g. from $z\mapsto 0$ or $z\mapsto e^z$, is infinitely differentiable in this set.

Each of the elementary functions of Liouville and Ritt is infinitely differentiable within the elementary functions. That is because the elementary functions form a differential field that is closed under differentiation.

The following are two examples of elementary functions with an n-th derivative which is predictable.

Let $m$, $n$ $\in\mathbb{N}_{+}$.

Your elementary functions shall be infinitely often integrable within the elementary functions. That means, all $n$-th degree antiderivatives of this functions have to be infinitely often differentiable within the elementary functions.

a)

A first simple class of such elementary functions are the polynomials. The antiderivative of a polynomial is again a polynomial. Therefore each $n$-th degree antiderivative of a polynomial is a polynomial again.

b)

A second simple class of such functions, $f$, are the elementary functions having one $n$-th degree derivative that is $f$ again:

$$\frac{d^{n}f(x)}{dx^{n}}=f(x)$$

Because the $n$-th degree antiderivative is an elementary function, all $m$-th degree antiderivatives with $m<n$ are elementary functions again. These differential equations can be easily solved.

$ $

Also the Liouvillain functions form a differential field that is closed under differentiation. Therefore they are also infinitely differentiable in the Liouvillian functions. The elementary functions are a subfield of the Liouvillian functions.

Each function from a set which can be generated by successive integration from a self-differentiable function $x\mapsto ce^x$, $c$ a constant, is infinitely differentiable in this set.

Each elementary function is infinitely differentiable within the elementary functions.

Let $m$, $n$ $\in\mathbb{N}_{+}$.

Your elementary functions shall be infinitely often integrable within the elementary functions. That means, all $n$-th degree antiderivatives of this functions have to be infinitely often differentiable within the elementary functions.

A first simple class of such functions are the polynomials. The antiderivative of a polynomial is again a polynomial. Therefore each $n$-th degree antiderivative of a polynomial is a polynomial again.

A second simple class of such elementary functions, $f$, are the elementary functions having one $n$-th degree derivative that is $f$ again:

$$\frac{d^{n}f(x)}{dx^{n}}=f(x)$$

Because the $n$-th degree antiderivative is an elementary function, all $m$-th degree antiderivatives with $m<n$ are elementary functions again. These differential equations can be easily solved.

Each of the elementary functions of Liouville and Ritt is infinitely differentiable within the elementary functions. That is because the elementary functions form a differential field that is closed under differentiation.

The following are two examples of elementary functions with an n-th derivative which is predictable.

Let $m$, $n$ $\in\mathbb{N}_{+}$.

Your elementary functions shall be infinitely often integrable within the elementary functions. That means, all $n$-th degree antiderivatives of this functions have to be infinitely often differentiable within the elementary functions.

a)

A first simple class of such elementary functions are the polynomials. The antiderivative of a polynomial is again a polynomial. Therefore each $n$-th degree antiderivative of a polynomial is a polynomial again.

b)

A second simple class of such elementary functions, $f$, are the elementary functions having one $n$-th degree derivative that is $f$ again:

$$\frac{d^{n}f(x)}{dx^{n}}=f(x)$$

Because the $n$-th degree antiderivative is an elementary function, all $m$-th degree antiderivatives with $m<n$ are elementary functions again. These differential equations can be easily solved.

$ $

Also the Liouvillain functions form a differential field that is closed under differentiation. Therefore they are also infinetly differentiable in the Liouvillian functions. The elementary functions are a subfield of the Liouvillian functions.

Each function from a set which can be generated by successive integration from a self-differentiable function, e.g. from $z\mapsto 0$ or $z\mapsto e^z$, is infinitely differentiable in this set.

Let $n\in\mathbb{N}_{0}$.

Your elementary functions shall be infinitely often integrable within the elementary functions. That means, all $n$-th degree antiderivatives of this functions have to be infinitely often differentiable within the elementary functions.

A first simple class of such functions are the polynomials. The antiderivative of a polynomial is again a polynomial. Therefore each $n$-th degree antiderivative of a polynomial is again a polynomial.

A second simple class of such elementary functions, $f$, are the elementary functions having an $n$-th degree derivative that is again $f$:

$$\frac{d^{n}f(x)}{dx^{n}}=f(x)$$

These differential equations can be easily solved.

Each elementary function is infinitely differentiable within the elementary functions.

Let $m$, $n$ $\in\mathbb{N}_{+}$.

Your elementary functions shall be infinitely often integrable within the elementary functions. That means, all $n$-th degree antiderivatives of this functions have to be infinitely often differentiable within the elementary functions.

A first simple class of such functions are the polynomials. The antiderivative of a polynomial is again a polynomial. Therefore each $n$-th degree antiderivative of a polynomial is a polynomial again.

A second simple class of such elementary functions, $f$, are the elementary functions having one $n$-th degree derivative that is $f$ again:

$$\frac{d^{n}f(x)}{dx^{n}}=f(x)$$

Because the $n$-th degree antiderivative is an elementary function, all $m$-th degree antiderivatives with $m<n$ are elementary functions again. These differential equations can be easily solved.