In analytic number theory it is traditional to write a complex variable as $s = \sigma + it$, with the letter $t$ going back to Riemann's paper on the zeta-function (1859) and the letter $\sigma$ having a more complicated explanation that was asked about on Mathoverflow here.

It recently came to my attention that in electrical engineering the notation $s = \sigma + j\omega$ is widely used, where $\omega$ describes a frequency. Engineers don't want to use $t$ as the imaginary part since $t$ always means time, and $j$ is a commonly used alternative to $i$ for $\sqrt{-1}$ since $i$ in electrical engineering often means current. My question is: who first introduced the notation $s = \sigma + j\omega$? I am wondering in particular if this use of $\sigma$ can be directly tied to the use of $\sigma$ as the real part of $s$ in analytic number theory. Edit: I am not wondering about the use of $j$ rather than $i$, so please don’t focus on that or refer me to this page, which does not answer my question.

One of the settings where the notation $s = \sigma + j\omega$ is used is in Bode plots, and I looked at several papers of Bode from the 1930s. In none of them did I find $s$ or $\sigma$ being used to describe complex numbers and their real parts. Complex numbers were introduced to electrical engineering by Charles Steinmetz, and in his 1893 paper "Complex Quantities and Their Use in Electrical Engineering" he introduced $j$ as $\sqrt{-1}$ and $\tilde{\omega}$ as a phase angle, but does not use $s$ or $\sigma$ in the above sense. He writes $s$ and $\sigma$ for completely different purposes: $s$ denotes inductance (a real quantity) and $\sigma$ denotes the EMF (electromotive force) of self-induction as a fraction of total EMF.

In analytic number theory it is traditional to write a complex variable as $s = \sigma + it$, with the letter $t$ going back to Riemann's paper on the zeta-function (1859) and the letter $\sigma$ having a more complicated explanation that was asked about on Mathoverflow here.

It recently came to my attention that in electrical engineering the notation $s = \sigma + j\omega$ is widely used, where $\omega$ describes a frequency. Engineers don't want to use $t$ as the imaginary part since $t$ always means time, and $j$ is a commonly used alternative to $i$ for $\sqrt{-1}$ since $i$ in electrical engineering often means current. My question is: who first introduced the notation $s = \sigma + j\omega$? I am wondering in particular if this use of $\sigma$ can be directly tied to the use of $\sigma$ as the real part of $s$ in analytic number theory. Edit: I am not wondering about the use of $j$ rather than $i$, so please don’t focus on that or refer me to this page, which does not answer my question.

One of the settings where the notation $s = \sigma + j\omega$ is used is in Bode plots, and I looked at several papers of Bode from the 1930s. In none of them did I find $s$ or $\sigma$ being used to describe complex numbers and their real parts. Complex numbers were introduced to electrical engineering by Charles Steinmetz, and in his 1893 paper "Complex Quantities and Their Use in Electrical Engineering" he introduced $j$ as $\sqrt{-1}$ and $\tilde{\omega}$ as a phase angle, but does not use $s$ or $\sigma$ in the above sense. He writes $s$ and $\sigma$ for completely different purposes: $s$ denotes inductance (a real quantity) and $\sigma$ denotes the EMF (electromotive force) of self-induction as a fraction of total EMF.

Note: This question is 1 day old and has 4 votes to close. I am baffled and would appreciate it if someone voting to close could please explain their reason for judging this question unsuitable to be asked here.

In analytic number theory it is traditional to write a complex variable as $s = \sigma + it$, with the letter $t$ going back to Riemann's paper on the zeta-function (1859) and the letter $\sigma$ having a more complicated explanation that was asked about on Mathoverflow here.

It recently came to my attention that in electrical engineering the notation $s = \sigma + j\omega$ is widely used, where $\omega$ describes a frequency. Engineers don't want to use $t$ as the imaginary part since $t$ always means time, and $j$ is a commonly used alternative to $i$ for $\sqrt{-1}$ since $i$ in electrical engineering often means current. My question is: who first introduced the notation $s = \sigma + j\omega$? I am wondering in particular if this use of $\sigma$ can be directly tied to the use of $\sigma$ as the real part of $s$ in analytic number theory.

One of the settings where the notation $s = \sigma + j\omega$ is used is in Bode plots, and I looked at several papers of Bode from the 1930s. In none of them did I find $s$ or $\sigma$ being used to describe complex numbers and their real parts. Complex numbers were introduced to electrical engineering by Charles Steinmetz, and in his 1893 paper "Complex Quantities and Their Use in Electrical Engineering" he introduced $j$ as $\sqrt{-1}$ and $\tilde{\omega}$ as a phase angle, but does not use $s$ or $\sigma$ in the above sense. He writes $s$ and $\sigma$ for completely different purposes: $s$ denotes inductance (a real quantity) and $\sigma$ denotes the EMF (electromotive force) of self-induction as a fraction of total EMF.

In analytic number theory it is traditional to write a complex variable as $s = \sigma + it$, with the letter $t$ going back to Riemann's paper on the zeta-function (1859) and the letter $\sigma$ having a more complicated explanation that was asked about on Mathoverflow here.

It recently came to my attention that in electrical engineering the notation $s = \sigma + j\omega$ is widely used, where $\omega$ describes a frequency. Engineers don't want to use $t$ as the imaginary part since $t$ always means time, and $j$ is a commonly used alternative to $i$ for $\sqrt{-1}$ since $i$ in electrical engineering often means current. My question is: who first introduced the notation $s = \sigma + j\omega$? I am wondering in particular if this use of $\sigma$ can be directly tied to the use of $\sigma$ as the real part of $s$ in analytic number theory. Edit: I am not wondering about the use of $j$ rather than $i$, so please don’t focus on that or refer me to this page, which does not answer my question.

One of the settings where the notation $s = \sigma + j\omega$ is used is in Bode plots, and I looked at several papers of Bode from the 1930s. In none of them did I find $s$ or $\sigma$ being used to describe complex numbers and their real parts. Complex numbers were introduced to electrical engineering by Charles Steinmetz, and in his 1893 paper "Complex Quantities and Their Use in Electrical Engineering" he introduced $j$ as $\sqrt{-1}$ and $\tilde{\omega}$ as a phase angle, but does not use $s$ or $\sigma$ in the above sense. He writes $s$ and $\sigma$ for completely different purposes: $s$ denotes inductance (a real quantity) and $\sigma$ denotes the EMF (electromotive force) of self-induction as a fraction of total EMF.