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The complex plane and Euler's formula

z = x + iy as a point, modulus and argument, e^{iθ} = cos θ + i sin θ.

Complex numbers are points in the plane; multiplication rotates and scales. Euler's formula makes rotation an exponential, so e^{iπ} = −1. Picture it: polar form of 1 + i — the point at distance √2 and angle 45°. Think it: the exponential map wraps the imaginary axis around the unit circle; that periodicity is why the complex logarithm has branches.

Exemplo trabalhado: e^(i*pi)

Arithmetic · Arithmetic

Evaluate e^(i*pi)

-1

Passo a passo

  1. e^{i \pi} = -1

    Power: E^(I·π) = -1.

Revelar a resposta
-1

Symbols used here

\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
i
imaginary unit
i² = −1.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\theta
theta
The usual name for an angle.
\frac{\partial f}{\partial x}
partial derivative
Derivative with respect to x, holding the other variables fixed.
\iint,\ \oint
double / contour integral
Integral over a region of the plane; integral around a closed curve.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\zeta(s)
Riemann zeta function
Σ 1/nˢ, continued to the complex plane; its zeros are the Riemann Hypothesis.
\operatorname{Res}_{z=a} f
residue
Coefficient of 1/(z − a) at a singularity; drives contour integrals.
f: \mathbb{C} \to \mathbb{C},\ \bar{z}
complex function, conjugate
A map of the complex plane; x − iy.

How to: The complex plane and Euler's formula

  1. Power: E^(I·π) = -1.

Questions people ask

Why is complex differentiability so much stronger than real?

The limit must be the same from every direction in the plane, not just two. That forces the Cauchy–Riemann equations, which in turn force infinitely many derivatives and a convergent Taylor series.

What is a residue?

The coefficient of 1/(z − a) in the Laurent series at a singularity a. The residue theorem says a contour integral equals 2πi times the sum of the residues inside, which evaluates many real integrals in one line.

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