In that case . . .

Colin Maclaurin gave sine, cosine and other trigonometric series.

Leonhard Euler must have discovered sine and cosine series himself and used them in e^i. theta.
If someone says Euler copied Maclaurin; I don't get disturbed.

I say that then Gauss is the king. Gauss seemed to be very high in favor of complex functions. Here is what he wrote. He wrote in German. Roughly translated in English he meant

"Complete knowledge of the nature of an analytic function must also include insight into its behavior for imaginary values of the arguments. Often the latter is indispensable even for a proper appreciation of the behavior of the function for real arguments. It is therefore essential that the original determination of the function concept be broadened to a domain of magnitudes which includes both the real and the imaginary quantities, on an equal footing, under the single designation complex numbers."

I believe in, "When your stomach is hungry, your hand reaches for food.".

Leonhard Euler is known for quality and quantity. His e^(i.pi)+1=0 is considered the best equation so far.

So, Leonhard Euler is the king of mathematics.

There is Euler-Maclaurin formula.

I believe in and too small in front of Reinhold Remmert. Reinhold Remmert says

"A.L. CAUCHY (1789-1857), B. RIEMANN (1826-1866), K. WEIERSTRASS (1815-1897). 

Each gave the theory a very distinct flavor and we still speak of the CAUCHY, the RIEMANN, and the WEIERSTRASS points of view. CAUCHY wrote his first works on function theory in the years 1814-1825. The function notion in use was that of his predecessors from the EULER era and was still quite inexact. To CAUCHY a holomorphic function was essentially a complex-differentiable function having a continuous derivative. CAUCHY's function theory is based on his famous integral theorem and on the residue concept. Every holomorphic function has a natural integral representation and is thereby accessible to the methods of analysis. The CAUCHY theory was completed by J. LIOUVILLE (1809-1882), [Liou]. The book [BB] of CH. BRIOT and J.-C. BOUQUET (1859) conveys a very good impression of the state of the theory at that time. Riemann's epochal Gottingen inaugural dissertation Grundlagen fair eine allgemeine Theorie der Functionen einer verdnderlichen complexen Grofle [R] appeared in 1851. To RIEMANN the geometric view was central: holomorphic functions are mappings between domains in the number plane C, or more generally between Riemann surfaces, "entsprechenden kleinsten Theilen ahnlich sind (correspondingly small parts of each of which are similar)." RIEMANN drew his ideas from, among other sources, intuition and experience in mathematical physics: the existence of current flows was proof enough for him that holomorphic (= conformal) mappings exist. He sought - with a minimum of calculation - to understand his functions, not by formulas but by means of the "intrinsic characteristic" properties, from which the extrinsic representation formulas necessarily arise. For WEIERSTRASS the point of departure was the power series; holomorphic functions are those which locally can be developed into convergent power series. Function theory is the theory of these series and is simply based in algebra. The beginnings of such a viewpoint go back to J.L. LAGRANGE. In his 1797 book Theorie des fonctions analytiques (2nd ed., Courcier, Paris 1813) he wanted to prove the proposition that every continuous function is developable into a power series. Since LAGRANGE HISTORICAL INTRODUCTION 5 we speak of analytic functions; at the same time it was supposed that these were precisely the functions which are useful in analysis. F. KLEIN writes "Die grofle Leistung von Weierstrafl ist es, die im Formalen stecken gebliebene Idee von Lagrange ausgebaut and vergeistigt zu haben (The great achievement of Weierstrass is to have animated and realized the program implicit in Lagrange's formulas)" (cf. p.254 of the German original of [H8]). And CARATHEODORY says in 1950 ([5], p.vii): WEIERSTRASS was able to "die Funktionentheorie arithmetisieren and ein System entwickeln, das an Strenge and Schonheit nicht iibertroffen werden kann (arithmetize function theory and develop a system of unsurpassable beauty and rigor)."