In two-dimensional fluid mechanics, specifically in the theory of the complex potential
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Taken together with the velocity potential, the stream function may be used to derive a complex potential.
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This complex potential was discussed as early as the nineteenth century, by the French mathematician Paul �mile Appell.
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From the physics of the problem it is deduced that the derivative of the complex potential w will look thus:
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Only one step is left to do : introduce w = f ( z ), the complex potential of the flow.
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If a circular cylinder | z | = a is placed into that flow, the complex potential for the new flow is given by
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Together, the potential function and the stream function form the complex potential, where the potential is the real part, and the stream function is the imaginary part.
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The equation describes the motion of a damped oscillator with a more complex potential than in simple harmonic motion ( which corresponds to the case \ beta = \ delta = 0 ); in physical terms, it models, for example, a spring pendulum whose spring's stiffness does not exactly obey Hooke's law.
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