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I have found many different claimed answers to this question:

Can somebody clarify this? Thanks a lot.

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d'Alembert did originally introduce the stream function in "Remarques sur les lois du mouvement des fluides" in 1761. Page 149 says that

$$p = \frac{\partial \omega}{\partial x} \,,$$

$$q = - \frac{\partial \omega}{\partial z} \,,$$

where $q = -u$, $p = -v$, $z = y$, and $\omega = \psi$ in modern notation. See page 138 for a better way to see how these variables match modern notation. This is the stream function once you parse the odd notation.

Lagrange basically repeated this more clearly in 1781 in "Mémoire sur la théorie du mouvement des fluides". See page 173 in particular. Lagrange changes the variables, though. He gives

$$p = \frac{\partial \omega}{\partial y} \,,$$

$$q = - \frac{\partial \omega}{\partial x} \,,$$

where $p = u$, $q = v$, and $\omega = \psi$. Other than that this is basically the modern notation for the stream function.

I'm not familiar with the history of the complex potential, so unfortunately I cannot help you there.

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    $\begingroup$ Amazing! Thanks a lot! My point was that if Euler really introduced the complex potential in 1757 it would mean that he also introduced the stream function, because it is just the imaginary part of the complex potential. It seems to me that one cannot really separate the history of the complex potential from the history of the stream function. $\endgroup$
    – timur
    Jan 2, 2022 at 16:16

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