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The 23 pages that link to Complex exponential, each with the reason it gives.

• Fourier transformRelated: Complex exponentials compactly encode the oscillations used as Fourier components.

• Fourier seriesCompared with: The complex form packages paired sine and cosine terms into single frequency components.

• Fourier analysisRelated: Complex exponentials provide compact notation for sinusoidal components.

• Periodic functionRelated: Complex exponentials provide compact building blocks for periodic oscillations.

• Euler's formulaNarrower topic: At imaginary inputs, its values trace the unit circle and yield this identity.

• TrigonometryCompared with: It packages trigonometric functions into a single expression and exposes their algebraic structure.

• Discrete Fourier transformRelated: The DFT compares samples with complex exponentials at its candidate frequencies.

• Entire functionBroader topic: It is a fundamental non-polynomial entire function with no zeros.

• Complex logarithmRelated: Its repeated values are exactly why each nonzero input has infinitely many logarithms.

• CosineRelated: Euler’s formula shows cosine as the real part of a complex rotation.

• Harmonic analysisRelated: Complex exponentials combine sine and cosine oscillations into one convenient basis.

• Plane waveRelated: Its oscillatory form expresses a plane wave’s phase dependence on position and time.

• Winding numberRelated: Its periodicity makes the net angle change around a loop an integer multiple of 2π.

• Lindemann–Weierstrass theoremRelated: The theorem concerns exponential values at algebraic inputs, including nonreal ones.

• Fourier coefficientRelated: Complex exponentials provide a compact basis whose expansion weights are Fourier coefficients.

• Euler's identityBroader topic: At input iπ, its magnitude is one and its angle is π.

• Complex derivativeBroader topic: Its complex derivative equals the function itself at every complex number.

• Polar formRelated: Its magnitude-and-phase form is the exponential notation for polar form.

• Jordan's lemmaRelated: Its magnitude on a semicircle creates the angular decay used by the lemma.

• Sine and cosineCompared with: Euler’s formula relates it to sine and cosine, while its exponential behavior extends beyond real angles.

• Four exponentials conjectureRelated: Each quantity tested by the conjecture is an exponential of a product.

• Hyperbolic angleCompared with: Circular angles arise from imaginary exponents, while hyperbolic angles arise from real exponents.

• Carlson's theoremRelated: Its behavior along complex directions gives exponential type its scale.