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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
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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