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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · e^(i*pi)

Arithmetic · Arithmetic

Evaluate e^(i*pi)

e^{i \pi}

Àwọn ìṣàmúlò-ètò

e^{i \pi} = -1
Power: E^(I·π) = -1.

Fi àwọn àgbèwọlé hàn

-1

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Bii o ṣe le: The complex plane and Euler's formula

Power: E^(I·π) = -1.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Complex Analysis

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