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  2. Euler's formula - Wikipedia

    en.wikipedia.org/wiki/Euler's_formula

    A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers.

  3. Pythagorean addition - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_addition

    This operation can be used in the conversion of Cartesian coordinates to polar coordinates. It also provides a simple notation and terminology for some formulas when its summands are complicated; for example, the energy-momentum relation in physics becomes E = m c 2 ⊕ p c . {\displaystyle E=mc^{2}\oplus pc.}

  4. Polar coordinate system - Wikipedia

    en.wikipedia.org/wiki/Polar_coordinate_system

    Similarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angle). Therefore, the same point ( r , φ ) can be expressed with an infinite number of different polar coordinates ( r , φ + n × 360°) and (− r , φ + 180° + n × 360°) = (− r , φ ...

  5. Complex conjugate - Wikipedia

    en.wikipedia.org/wiki/Complex_conjugate

    In polar form, if and are real numbers then the conjugate of is . This can be shown using Euler's formula . The product of a complex number and its conjugate is a real number: a 2 + b 2 {\displaystyle a^{2}+b^{2}} (or r 2 {\displaystyle r^{2}} in polar coordinates ).

  6. Complex plane - Wikipedia

    en.wikipedia.org/wiki/Complex_plane

    The multiplication of two complex numbers can be expressed more easily in polar coordinates: the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a ...

  7. Coordinate systems for the hyperbolic plane - Wikipedia

    en.wikipedia.org/wiki/Coordinate_systems_for_the...

    The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, or polar angle.

  8. Change of variables - Wikipedia

    en.wikipedia.org/wiki/Change_of_variables

    Difficult integrals may also be solved by simplifying the integral using a change of variables given by the corresponding Jacobian matrix and determinant. [1] Using the Jacobian determinant and the corresponding change of variable that it gives is the basis of coordinate systems such as polar, cylindrical, and spherical coordinate systems.

  9. In-phase and quadrature components - Wikipedia

    en.wikipedia.org/wiki/In-phase_and_quadrature...

    In vector analysis, a vector with polar coordinates A, φ and Cartesian coordinates x = A cos(φ), y = A sin(φ), can be represented as the sum of orthogonal components: [x, 0] + [0, y]. Similarly in trigonometry, the angle sum identity expresses: