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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.

Worked example · e^(i*pi)

Arithmetic · Arithmetic

Evaluate e^(i*pi)

e^{i \pi}

Step by step

e^{i \pi} = -1
Power: E^(I·π) = -1.

Reveal the answer

-1

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How to: The complex plane and Euler's formula

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.

More in Complex Analysis

Holomorphic functions and the Cauchy-Riemann equations Contour integrals and the residue theorem Power series and analytic continuation