In fact, Alex Wilkie showed that the situation is even worse: he constructed a transcendental function ƒ: R → R that is analytic everywhere but whose transcendence cannot be detected by any first-order method.
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Some systems (particularly older, microcode-based architectures) can also perform various transcendental functions such as exponential or trigonometric calculations, though in most modern processors these are done with software library routines.
This became a topic of his Master's thesis titled "On the expansion of transcendental function in partial fractions. After publishing his thesis Bukreev went abroad and took lectures of Karl Weierstrass, Lazarus Fuchs, and Leopold Kronecker in Berlin. Bukreev undertook research on Fuchsian functions under Fuchs' guidance, which he completed in 1888 and which became the basis of his doctoral thesis "On the Fuchsian functions of rank zero" defended in 1889.