Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-20T04:23:33.100Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Functional Equations f(x+iy)| = |f(x)+f(iy)| and |f(x+iy)| = |f(x)-f(iy)| and on Ivory's Theorem

Published online by Cambridge University Press:  20 November 2018

Hiroshi Haruki
Hiroshi Haruki*
Affiliation:
Mathematical Institute, Osaka University, Osaka, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Considering Cauchy's functional equation

f(z1+z2)=f(z1)+ f(z2),

where f(z) is an entire function of z, we have the following functional equation:

(1) |f(x+iy)|=|f(x)+f(iy)|,

where x and y are real.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 9 , Issue 4 , October 1966 , pp. 473 - 480
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Robinson, R.M., A Curious Trigonometric Identity. Amer. Math. Monthly 64, (1957), pages 83-85.Google Scholar
2. Haruki, H., On Ivory′s Theorem. Mathematica Japonicae, Vol. 1, No. 4, page 151, (1949).Google Scholar
3. Haruki, H., Studies on Certain Functional Equations from the Standpoint of Analytic Function Theory. Sci. Rep. College of General Education, Osaka Univ., Vol. 14, No. 1, page 32, (1965).Google Scholar