Introduction
Hello everyone! This is my blog about Maths. Posts will be daily with probability ε, for ε sufficiently small ;) I will also talk about interesting Maths resources and websites.
An interesting problem
Here is an interesting functional equation problem. Can you solve it? Find all functions so that a) for all and b) for all Here is how I solved it. Substitute u for -u and v for -v. Then our functional equation becomes F(-2u)= F(u+v)F(v-u)+F(u-v)F(-u-v)=F(2u). Let 2u=x, then we get F(x)=F(-x). Now let u=0. So we get F(0)=F(v)^2+ F(v)F(-v)= 2F(v)^2. If we let v=0, we get that F(0)=2F(0)^2. Solving for F(0), we get that F(0)= 0 or F(0)= 1/2. If F(0)=0, then F(x)=0 for all x. But if F(0)= 1/2, then F(x)= 1/2 or -1/2 for all x. However, F(x)≥0, so the only solutions are F(x)=0 or F(x)= 1/2.
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