[Math] Writing a non integer power in terms of integer powers

exponentiation

I would like to write $x^{2.5}$ in terms of $x$ to the power of integers, is there any way to do this. Taylor series etc. don't work when they depend on derivatives.

If it is not possible, do you have or know a proof.

Thanks

EDIT:

to clarify, I mean that I want to write $x^{2.5}$ in terms of a series of $x^{\mathrm{integer}}$'s, for example: $1 + x^2 + x^{3}$.
I tried to use Taylor series but since it depends on derivatives of $x^{2.5}$ but they do not have integer powers…

Best Answer

$x^{2.5}$ can't be represented by a series in integer powers of $x$, because it is not analytic (in fact not meromorphic) at $0$: instead, it has a branch point there.

Related Solutions

[Math] Algorithm for computing powers

squared exponentiation is the way to go here

BigInt pow(BigInt a,BigInt n){
    BigInt r= 1;
    while(n>0){
        if(mod(n,2)==1)r=mult(r,a);
        a=mult(a,a);
        n=div(n,2);
    }
    return r;
}

here mod(a,b) is the remainder of a/b mult(a,b) is a*b and div(a,b) is floor(a/b)

this relies on $a^n = {a^2}^{\lfloor \frac{n}{2}\rfloor }\cdot \begin{cases} a & \mbox{if }n\mbox{ is even}\\1 & \mbox{if }n\mbox{ is odd}\end{cases}$

or more precisely

$$a^n=\prod_{i=0}^{\lfloor\log n\rfloor}{\begin{cases} a^{2^i} & \mbox{if }\Big\lfloor \frac{n}{2^i}\Big\rfloor\mbox{ is even}\\1 & \mbox{if }\Big\lfloor \frac{n}{2^i}\Big\rfloor\mbox{ is odd}\end{cases}} $$

[Math] Non-integral powers of a matrix

As @tom pointed out in a comment, the power of a matrix can be defined in terms of logarithm of a matrix and matrix exponential, using

$$A^p:=\exp\left(p\ln A\right)$$

Using the principal logarithm (this is the name for that choice described in the question without giving a name), the above even yields unique results.

The matrix exponential is defined for every matrix, the matrix logarithm only for invertible matrices. The case of singular matrices mentioned in the question is therefore not covered by this definition.

EDIT:

Best Answer

Related Solutions

[Math] Algorithm for computing powers

[Math] Non-integral powers of a matrix

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