- #1

- 1,452

- 9

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- I
- Thread starter friend
- Start date

- #1

- 1,452

- 9

- #2

mathwonk

Science Advisor

Homework Helper

- 11,306

- 1,521

try the infinite series definition of exponential. what happens?

- #3

- 1,452

- 9

The Taylor series for the exponential is,try the infinite series definition of exponential. what happens?

[itex]{e^x} = 1 + \frac{x}{{1!}} + \frac{{{x^2}}}{{2!}} + \frac{{{x^3}}}{{3!}} + \frac{{{x^4}}}{{4!}} + ...[/itex]

So obviously the linear approximation for the exponential will be the same as the exponent. If the exponent is a quaternion, then the linear approximation will be a quaternion, and same goes for the octonions. However, the quaternions and octonions do not commute. I think this means that you can still get a quadratic approximation (correct me if I'm wrong), but odd numbered exponents will have two ways of being expressed, depending on whether you multiply from the left or right. So I don't know what to say about this.

- #4

- 1,452

- 9

[itex]q = {x_0} + i{x_1} + j{x_2} + k{x_3}[/itex],

where [itex]{x_0}[/itex], [itex]{x_1}[/itex], [itex]{x_2}[/itex], [itex]{x_3}[/itex] are real numbers. And [itex]{i^2} = {j^2} = {k^2} = - 1[/itex]

Its conjugation is written,

[itex]{q^*} = {x_0} - i{x_1} - j{x_2} - k{x_3}[/itex].

Its norm can be written,

[itex]\left\| q \right\| = \sqrt {q{q^*}} = \sqrt {{x_0}^2 + {x_1}^2 + {x_2}^2 + {x_3}^2} [/itex].

But

[itex]q = {x_0} + \vec v[/itex],

where,

[itex]\vec v = i{x_1} + j{x_2} + k{x_3}[/itex].

Then the exponential of a quaternion,

[itex]{e^q} = \sum\limits_{n = 0}^\infty {\frac{{{q^n}}}{{n!}} = {e^{{x_0}}}(\cos \left\| {\vec v} \right\|} + \frac{{\vec v}}{{\left\| {\vec v} \right\|}}\sin \left\| {\vec v} \right\|)[/itex].

So since the [itex]{\vec v}[/itex] carries the

I assume the same construction leads to the fact that the exponential of a octonion is an octonion.

Share: