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

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DOI

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
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     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/}
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