Home Page | Overview | Site Map | Index | Appendix | Illustration | About | Contact | Update | FAQ |
Let us start from the differentiation of the logarithmic function loga(y) with an arbitrary base a: d[loga(y)] loga(y + y) - loga(y) loga(1 + y/y) (y/y) loga(1 + y/y)(y/y) ---------- (30) where we have applied the logarithmic relations (Figure 11): loga(u/v) = loga(u) - loga(v) ---------- (31a) loga(u)n = n loga(u) ---------- (31b). | |
Figure 11 Logarithm |
y = ei x = cos x + i sin x ---------- (34) This relationship was discovered by the Swiss mathematician Leonhard Euler (1707-83). The modern derivation can be obtained easily if we perform a Taylor series expansion about the point xo = 0 for ex, substituting to the independent variable with ix and comparing the result with Eqs.(29a,b). The Euler's divine formula (Figure 13) : ei + 1 = 0 ---------- (35) | ||
Figure 12 Log-a-Rhythm [view large image] |
Figure 13 Euler's Formula [view large image] |
is a straight forward application of Eq.(34). It contains all the special symbols in mathematics: e, i = , , 1, and 0. |