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So I've been told that you can't find the closed form of $\int e^{-\frac{x^2}{2}}$.

Apparently, you know the exact result then you integrate over the whole of $\mathbb{R}$ but every other number needs to be calculated numerically.

What I'm wondering is why is this considered worse (or maybe it isn't?) than say, $\int\cos x=\sin x$.

As far as I'm aware, you can calculate the exact value of $\sin$ for a very limited number of points and the rest are again calculated numerically. So any not just give a special name to the first integral, or maybe something more elementary, and say that's also a closed form?

What makes this different from trigonometric functions?

Edit: I'm also a bit fuzzy on why/if getting $\pi$ as a result is considered exact. It's just a name for something that we can't express in a closed form, right?

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Why is this considered worse $($or maybe it isn't ?$)$ than say $\displaystyle\int\cos x~dx=\sin x$ ?

Because all integrals whose integrand consists solely of trigonometric functions $($whose arguments are integer multiples of $x)$, in combination with the four basic operations and exponentiation to an integer power, can be expressed in closed form using the Weierstrass substitution, followed by partial fraction decomposition. $\big($Of course, certain algebraic constants might appear there, which are not expressible in radicals, but this is another story$\big)$. In other words, they don't create anything new, since $\cos x=\sin\bigg(x+\dfrac\pi2\bigg)$. But the indefinite integral of a Gaussian function does create something new, namely the error function, and then, when we further integrate that, we get something even newer $\bigg($since $\displaystyle\int\text{erf}(x)^3~dx$ cannot be expressed as a combination of elementary and error functions$)$, etc. and it just never stops. So trigonometric, hyperbolic, and exponential functions are self-contained under integration, in a way in which Gaussian ones simply aren't.

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You raise a good point here, and you're intuition is on the mark. In fact, the indefinite integral you gave is sometimes given the name $\text{erf}$ much like saying $\int\cos x=\sin x$.

This probably why you don't see a rigorous definition of closed form.

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