Geodesic
On the Functional Equations f(x+iy)| = |f(x)+f(iy)| and |f(x+iy)| = |f(x)-f(iy)| and on Ivory's Theorem
Canadian mathematical bulletin, Tome 9 (1966) no. 4, pp. 473-480

Voir la notice de l'article provenant de la source Cambridge University Press

Considering Cauchy's functional equationf(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. 
Haruki, Hiroshi. On the Functional Equations f(x+iy)| = |f(x)+f(iy)| and |f(x+iy)| = |f(x)-f(iy)| and on Ivory's Theorem. Canadian mathematical bulletin, Tome 9 (1966) no. 4, pp. 473-480. doi: 10.4153/CMB-1966-058-x
@article{10_4153_CMB_1966_058_x,
     author = {Haruki, Hiroshi},
     title = {On the {Functional} {Equations} f(x+iy)| = |f(x)+f(iy)| and |f(x+iy)| = |f(x)-f(iy)| and on {Ivory's} {Theorem}},
     journal = {Canadian mathematical bulletin},
     pages = {473--480},
     year = {1966},
     volume = {9},
     number = {4},
     doi = {10.4153/CMB-1966-058-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-058-x/}
}
TY  - JOUR
AU  - Haruki, Hiroshi
TI  - On the Functional Equations f(x+iy)| = |f(x)+f(iy)| and |f(x+iy)| = |f(x)-f(iy)| and on Ivory's Theorem
JO  - Canadian mathematical bulletin
PY  - 1966
SP  - 473
EP  - 480
VL  - 9
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-058-x/
DO  - 10.4153/CMB-1966-058-x
ID  - 10_4153_CMB_1966_058_x
ER  - 
%0 Journal Article
%A Haruki, Hiroshi
%T On the Functional Equations f(x+iy)| = |f(x)+f(iy)| and |f(x+iy)| = |f(x)-f(iy)| and on Ivory's Theorem
%J Canadian mathematical bulletin
%D 1966
%P 473-480
%V 9
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-058-x/
%R 10.4153/CMB-1966-058-x
%F 10_4153_CMB_1966_058_x

[1] 1. Robinson, R.M., A Curious Trigonometric Identity. Amer. Math. Monthly 64, (1957), pages 83-85. Google Scholar

[2] 2. Haruki, H., On Ivory′s Theorem. Mathematica Japonicae, Vol. 1, No. 4, page 151, (1949). Google Scholar

[3] 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

Cité par Sources :