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3.1

Show that

$$ tanh(a)=2 \sigma (2a) - 1 $$

Then show

$$ y(x, \boldsymbol w) = w_0 + \Sigma_{j=1}^M w_j \sigma ( \frac{x - \mu_j}{s} ) $$

is equivalent to a linear combination of 'tanh' functions of the form

$$ y(x, \boldsymbol u) = w_0 + \Sigma_{j=1}^M u_j tanh ( \frac{x - \mu_j}{s} ) $$

$$ tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} $$$$ \sigma(x) = \frac{1}{1 + e^{-x}} $$

$$ 2\sigma(2a) - 1 = \frac{2}{1 + e^{-2a}} - 1 $$

$$ =\frac{1 - e^{-2a}}{1 + e^{-2a}} $$

$$ = \frac{e^a - e^{-a}}{e^a + e^{-a}} = tanh(a) $$

solve for

$$ w * \sigma(z) = u * tanh(z) $$

$$ w * \sigma (z) = u * ( 2 \sigma(2z) - 1 ) $$

$$ u = \frac{w\sigma(z)}{2\sigma(2z) - 1} $$