In the late nineteenth century, Riemann proposed the function as an example of a continuous function that is almost nowhere differentiable. This Demonstration shows the graphs of generalized versions of Riemann's function, defined by , where and are real parameters satisfying and , and corresponding functions are obtained by replacing by or .

Given two real parameters and , the generalized complex Riemann function is defined as

.

The generalized Riemann sine and cosine functions are defined as the imaginary and real parts of , respectively:

,

All three types of Riemann functions are known to be continuous but almost nowhere differentiable, and their graphs exhibit fractal behavior [1–3].

This Demonstration shows the graphs of the real-valued Riemann sine and cosine functions, and , over the interval , and the graph of the complex Riemann function, , , represented by a path in the complex plane. In the case when is an integer, the function is periodic with period 1, so the corresponding path is closed.

References

[1] F. Chamizo and A. Córdoba, "Differentiability and Dimension of Some Fractal Fourier Series," Advances in Mathematics, 142(2), 1999 pp. 335–354. doi:10.1006/aima.1998.1792.

[2] J. J. Duistermaat, "Selfsimilarity of 'Riemann's Nondifferentiable Function'," Nieuw Archief voor Wiskunde, 9(3), 1991 pp. 303–337. https://dspace.library.uu.nl/handle/1874/2601.

[3] J. Gerver, "More on the Differentiability of the Riemann Function," American Journal of Mathematics, 93(1), 1971 pp. 33–41. doi:10.2307/2373445.