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  2. Complex number - Wikipedia

    en.wikipedia.org/wiki/Complex_number

    A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.

  3. Lagrange's identity - Wikipedia

    en.wikipedia.org/wiki/Lagrange's_identity

    Lagrange's identity for complex numbers has been obtained from a straightforward product identity. A derivation for the reals is obviously even more succinct. Since the Cauchy–Schwarz inequality is a particular case of Lagrange's identity, [4] this proof is yet another way to obtain the CS inequality. Higher order terms in the series produce ...

  4. cis (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Cis_(mathematics)

    x is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula , e ix , which offers an even shorter notation for cos x + i sin x , but cis(x) is widely used as a name for this function in software libraries .

  5. Complex conjugate - Wikipedia

    en.wikipedia.org/wiki/Complex_conjugate

    In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}

  6. Euler's formula - Wikipedia

    en.wikipedia.org/wiki/Euler's_formula

    In fact, the same proof shows that Euler's formula is even valid for all complex numbers x. 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 ...

  7. Residue (complex analysis) - Wikipedia

    en.wikipedia.org/wiki/Residue_(complex_analysis)

    Suppose a punctured disk D = {z : 0 < |z − c| < R} in the complex plane is given and f is a holomorphic function defined (at least) on D. The residue Res( f , c ) of f at c is the coefficient a −1 of ( z − c ) −1 in the Laurent series expansion of f around c .

  8. Ramanujan's master theorem - Wikipedia

    en.wikipedia.org/wiki/Ramanujan's_master_theorem

    The number of brackets is the number of linear equations associated with an integral. This term reflects the common practice of bracketing each linear equation. [15] The complexity index is the number of integrand sums minus the number of brackets (linear equations). Each series expansion of the integrand contributes one sum.

  9. Cayley–Dickson construction - Wikipedia

    en.wikipedia.org/wiki/Cayley–Dickson_construction

    As a complex number consists of two independent real numbers, they form a two-dimensional vector space over the real numbers. Besides being of higher dimension, the complex numbers can be said to lack one algebraic property of the real numbers: a real number is its own conjugate.