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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.

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    \$\begingroup\$ All the formulas you wrote are equivalent (except the third which is incorrect). \$\endgroup\$
    – Mat
    Dec 27 '21 at 16:24
  • \$\begingroup\$ 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? \$\endgroup\$
    – Nava Moore
    Dec 27 '21 at 16:30
  • \$\begingroup\$ Specifically where is your confusion? \$\endgroup\$
    – Andy aka
    Dec 27 '21 at 17:44
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    \$\begingroup\$ @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. \$\endgroup\$
    – jonk
    Dec 27 '21 at 18:47
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    \$\begingroup\$ Slight correction @jonk: the imaginary part of the complex frequency \$s\$ should be \$\omega\$, not \$\omega t\$. \$\endgroup\$
    – alejnavab
    Dec 27 '21 at 20:43
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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:

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

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