f(x)=arccosh(x)=log(x+(x^2-1)^1/2)

f(t)=arccosh(3t/2)=log(3t/2+1/2・(9t^2-4)^1/2)=log(3t+(9t^2-4)^1/2)-log2

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f(x)+f(y)=log(3x+(9x^2-4)^1/2)(3y+(9y^2-4)^1/2)-2log2

f(z)=log(3z+(9z^2-4)^1/2)-log2

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もし、z^2-3xyz+x^2+y^2-4/9=0

z=1/2・{3xy+(9x^2y^2-4x^2-4y^2+16/9)^1/2}

z=1/2・{3xy+1/3・(81x^2y^2-36x^2-36y^2+16)^1/2}

z=1/2・{3xy+1/3・{(9x^2-4)(9y^2-4)}^1/2}

であれば、

3z=1/2・{9xy+{(9x^2-4)(9y^2-4)}^1/2}

9z^2-4=1/4・{81x^2y^2+{(9x^2-4)(9y^2-4)}+18xy{(9x^2-4)(9y^2-4)}^1/2-16}

9z^2-4=1/4・{81x^2y^2+81x^2y^2-36x^2-36y^2+16+18xy{(9x^2-4)(9y^2-4)}^1/2-16}

9z^2-4=1/4・{9x^2(9y^2-4)+9y^2(9x^2-4)+18xy{(9x^2-4)(9y^2-4)}^1/2}

9z^2-4=1/4・{3x(9y^2-4)^1/2+3y(9x^2-4)^1/2}^2

3z+(9z^2-4)^1/2=1/2・{9xy+{(9x^2-4)(9y^2-4)}^1/2}+1/2・{3x(9y^2-4)^1/2+3y(9x^2-4)^1/2}

=1/2・(3x+(9x^2-4)^1/2)(3y+(9y^2-4)^1/2

f(z)=log(3z+(9z^2-4)^1/2)-log2=f(x)+f(y)

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