The Kelvin functions (sometimes known as the Thompson functions) are defined by the following equations:

The Kelvin functions are all real valued for real x and positive v.

Abramowitz, M., and Stegun, I. Handbook of Mathematical Functions, Section 9.9. Washington: National Bureau of Standards Applied Mathematics, 1964.

Erdelyi, A., ed. Higher Transcendental Functions, Section 7.2.3. New York: McGraw-Hill, 1953.