# Understanding Euler's formula in different forms

I am a little confused on the form of Euler's formula.

I'm taking a signal and signals class where the book uses Euler's formula as:

$$e^{j\theta} = cos\theta +jsin\theta$$ $$e^{-j\theta}=cos\theta -jsin\theta$$

In my circuit analysis class my professor wrote Euler's formula as:

$$e^{-\sigma t}e^{-j\omega t} = cos\omega t -jsin\omega t$$ $$e^{-\sigma t}e^{-j\omega t}=e^{-\sigma t}cos\omega t -je^{-\sigma t}sin\omega t$$

$$e^{j\omega t}=cos\omega t +jsin\omega t$$

I'm confused on what the general formula should be and understanding how and why the form of the equation changes.

Any clarification on this is greatly appreciated.

• All the formulas you wrote are equivalent (except the third which is incorrect).
– Mat
Dec 27 '21 at 16:24
• Thank you. I think I corrected it from my notes. If they are all equivalent, could you please explain why they are used in those forms? Dec 27 '21 at 16:30
• Specifically where is your confusion? Dec 27 '21 at 17:44
• @NavaMoore s-space is widely used in electronics. In math, similar ideas are involved in complex analysis. In electronics, $s=\sigma+j\,\omega t$ and is a complex number. When you write $e^{^{s t}}$ this is the same thing as writing $e^{^{\sigma t}}e^{^{j\,\omega t}}$. There's no difference. You can use Euler's to expand the second factor, of course. The meaning of these, in the context of time, is that the first factor converges towards 0 for $\sigma\lt 0$, is a constant 1 when $\sigma= 0$ and diverges when $\sigma\gt 0$. In electronics, $\sigma\gt 0$ is usually a bad thing.
– jonk
Dec 27 '21 at 18:47
• Slight correction @jonk: the imaginary part of the complex frequency $s$ should be $\omega$, not $\omega t$. Dec 27 '21 at 20:43

Your third equation is only correct, if and only if $$\\sigma=0\$$.

Well, when we state Euler's formula $$\\forall\space x\in\mathbb{R}\$$:

$$\exp\left(x\text{j}\right)=\cos\left(x\right)+\text{j}\sin\left(x\right)\tag1$$

Were $$\\text{j}^2=-1\$$.

So, when we have:

$$\exp\left(-x\text{j}\right)=\cos\left(-x\right)+\text{j}\sin\left(-x\right)=\cos\left(x\right)-\text{j}\sin\left(x\right)\tag2$$

Because $$\\cos\left(\cdot\right)\$$ is an even function and $$\\sin\left(\cdot\right)\$$ is an odd function.

Now, when $$\t\space\wedge\space x\space\wedge\space\sigma\in\mathbb{R}\$$ we get:

$$\exp\left(-\sigma t\right)\exp\left(-x\text{j}\right)=\exp\left(-\sigma t\right)\left(\cos\left(x\right)-\text{j}\sin\left(x\right)\right)=$$ $$\exp\left(-\sigma t\right)\cos\left(x\right)-\text{j}\exp\left(-\sigma t\right)\sin\left(x\right)\tag3$$