On Brannan’s Conjecture

Abstract

We will prove the Brannan conjecture for particular values of the parameter. The basic tool of the study is an integral representation published in a recent work [3].

Complex number and its discovery history

“…Actually, this system is called "complex analysis". The development of complex numbers as an extension of rigorous proofs has provided new and expanding perspectives in many areas of mathematics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

“…The first traces of imaginary numbers were found in Italy, hidden in the cubic equation. In about 1530s, Nicolo Tartaglia, the mathematical genius, came on the scene [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Tartaglia often called a "stammerer" because of his speech impediment and the massacre that took place in his hometown as a child.…”

“…The use of method for cubic equations of a certain form was one of his primary achievements [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. His technique in his era was revolutionary.…”

“…In his rule, he defined x = t − u which is related to his cube, though Tartaglia's solution for z is still used. After Cardano geometrically solving his cube, he came up with a general solution [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

“…In some cases, like what is in Equation 1, if containing some negativities, it would present results with radicand being negative without validating them. Since the non-real value are computed by the 'rules of ordinary arithmetic', scientists have been trying to prove their existence for more than three hundred years until they finally realized what they are proving is only imaginary quantity [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. At that point, Cardano is not only trying to solve cubic equations in Tartaglia's version but also in a new form of parameterizing as 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑 = 0.…”