Real elementary functions and their domains

Every integral below is the oriented Riemann integral constructed from upper and lower Darboux sums in Continuous functions have an oriented Riemann integral. It exists for each displayed continuous integrand on the finite interval between its endpoints, is additive in intervals, reverses sign when the endpoints are reversed, is monotone in the integrand, and satisfies $\mathrm{\mid }{\int }_a^{b}g(t)dt\mathrm{\mid }\le \mathrm{\mid }b-a\mathrm{\mid }{\max }_{t\text{ between }a\text{ and }b}\mathrm{\mid }g(t)\mathrm{\mid }$. These properties also give constant comparison bounds and the average-value difference quotient used to differentiate the integral-defined functions.

The exponential is $\exp (x)={\sum }_{k=0}^{\mathrm{\infty }}{x}^k\mathrm{/}k!$ for $x\in \mathbb{R}$. The sine and cosine are the everywhere-convergent power series $\sin x={\sum }_{k=0}^{\mathrm{\infty }}(-1{)}^k{x}^{2k+1}\mathrm{/}(2k+1)!$ and $\cos x={\sum }_{k=0}^{\mathrm{\infty }}(-1{)}^k{x}^{2k}\mathrm{/}(2k)!$. Define $\arctan x={\int }_0^{x}dt\mathrm{/}(1+t^2)$ and set $\pi =4\arctan 1$. The natural logarithm $\ln$ is defined on $(0,\mathrm{\infty })$ by $\ln x={\int }_1^{x}dt\mathrm{/}t$.

The square root is the unique nonnegative real $y$ with $y^2=x$ for $x\ge 0$; existence and uniqueness follow from Nonnegative real square roots exist uniquely. The tangent is $\tan x=\sin x\mathrm{/}\cos x$ on the set where $\cos x\mathrm{\ne }0$.

For complex $z=x+iy$, the modulus $\mathrm{\mid }z\mathrm{\mid }=\sqrt{x^2+y^2}$ is defined in Real intervals, complex rectangles and ball enclosures. The complex exponential is $\exp (x+iy)=\exp (x)(\cos y+i\sin y)$. The principal complex logarithm is $\mathrm{Log}(z)=\ln \mathrm{\mid }z\mathrm{\mid }+i\mathrm{Arg}(z)$ on $\mathbb{C}\setminus (-\mathrm{\infty },0]$. For $z=x+iy$ in this domain, define $\mathrm{Arg}(x+iy)=\{\begin{array}{ll}\arctan (y\mathrm{/}x),& x>0,\\ \pi \mathrm{/}2,& x=0,y>0,\\ -\pi \mathrm{/}2,& x=0,y<0,\\ \pi -\arctan (y\mathrm{/}(-x)),& x<0,y>0,\\ -\pi +\arctan ((-y)\mathrm{/}(-x)),& x<0,y<0.\end{array}$ The excluded negative real axis is the remaining case $x<0,y=0$; zero is also excluded.

These are point functions. An interval extension must separately enclose their exact ranges, test the stated domains, and return a non-finite status when its finite budget cannot certify a reduction or endpoint.