Logarithmic Integral Function

$\{\backslash rm\; li\}\; (x)\; =\; \backslash int\_\{0\}^\{x\}\; \backslash frac\{dt\}\{\backslash ln\; (t)\}\; \backslash ;\; .$

$\{\backslash rm\; li\}\; (x)\; =\; \backslash lim\_\{\backslash varepsilon\; \backslash to\; 0\}\; \backslash left(\; \backslash int\_\{0\}^\{1-\backslash varepsilon\}\; \backslash frac\{dt\}\{\backslash ln\; (t)\}\; +\; \backslash int\_\{1+\backslash varepsilon\}^\{x\}\; \backslash frac\{dt\}\{\backslash ln\; (t)\}\; \backslash right)\; \backslash ;\; .$

$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash frac\{M(t)\}\{1+t^2\}dt$.

$\{\backslash rm\; li\}\; (x)\; =\; \backslash Theta\; \backslash left(\; \{x\backslash over\; \backslash ln\; (x)\}\; \backslash right)\; \backslash ;\; .$

π(x) ~ li(x) ~ Li(x)

li(x) = Ei (ln (x))    for all positive real x ≠ 1.

$\{\backslash rm\; li\}\; (e^\{u\})\; =\; \backslash gamma\; +\; \backslash ln\; \backslash left|\; (u)\; \backslash right|\; +\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \{u^\{n\}\backslash over\; n\; \backslash cdot\; n!\}\; \backslash quad\; \{\backslash rm\; for\}\; \backslash ;\; u\; \backslash ne\; 0\; \backslash ;\; ,$

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