Hi,

In the plot below I have $\displaystyle f(\theta) = sin(\theta)$.

The second graph of the plot shows that, for x = 1, the plot of $\displaystyle P(\theta, x) = f(\theta - x f(\theta))$ has an infinite slope at $\displaystyle \theta = 0$ (called the Poisson non-linear propagation solution).

I'm trying to show this analytically but not getting the right answer. I reproduce my steps below.

We want to show that for $\displaystyle x = 1$, $\displaystyle P(\theta = 0, x)$ has an infinite slope, i.e. $\displaystyle dP/d\theta \rightarrow \inf$.

$\displaystyle P(\theta, x) = f(\theta - x f(\theta))$

we have $\displaystyle f(\theta) = sin(\theta)$, so

$\displaystyle P(\theta, x) = sin(\theta - x sin(\theta))$

$\displaystyle dP/d\theta = cos(\theta - sin(\theta))(1 - x cos(\theta))$

What I'm trying to show is $\displaystyle dP/d\theta \rightarrow \inf$, but that's clearly not possible from my result above.

Keen to hear where I'm going wrong. Thanks