Question

What is eulers identity derivation

Answer 1

Euler's identity is a mathematical equation that relates important constants. It can be derived by combining the power series expansions of exponential, cosine, and sine functions. The final equation is e^(iπ) + 1 = 0.

Step-by-step explanation:

Euler's identity is a mathematical equation that relates five important mathematical constants: e (the base of the natural logarithm), i (the imaginary unit), π (pi), 1 (the multiplicative identity), and 0 (the additive identity). The derivation of Euler's identity involves combining the power series expansions of the exponential function e^x, the cosine function cos(x), and the sine function sin(x) into a single equation.

To derive Euler's identity, we start with the power series expansions of e^ix, cos(x), and sin(x). By carefully manipulating these equations using properties of complex numbers, we arrive at the equation e^(ix) = cos(x) + i sin(x). This equation is known as Euler's formula, and when we substitute π for x, we get Euler's identity: e^(iπ) + 1 = 0.

Answer 2

Hello friend! Hope this helps!

Euler's formula is eⁱˣ=cos(x)+i⋅sin(x), and Euler's Identity is e^(iπ)+1=0. See how these are obtained from the Maclaurin series of cos(x), sin(x), and eˣ.