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  • ...ath> and lies on the unit circle. If any of the denominators are zero, the entire result follows trivially. Let <math>\mathbf{p} = \frac{1}{2}(\mathbf{u}+\ga *Consider the function <math>f(x)=\frac{(x+k)^2}{x^2+1},x\in (-\infty,\infty)</math>, where <math>
    13 KB (2,048 words) - 14:28, 22 February 2024
  • ...y]]! This is certainly not true of a real function, even a [[real analytic function]]. ...for some real number <math>A</math>, then <math>f</math> is a [[constant]] function.
    2 KB (271 words) - 21:06, 12 April 2022
  • ...orem]] of [[complex analysis]], which says that every [[bounded]] [[entire function]] is [[constant]]. ...at <math>P</math> is [[monic]]. Then <math>1/P(z)</math> is an [[entire]] function; we wish to show that it is bounded. It is clearly bounded when <math>n=0<
    5 KB (832 words) - 13:22, 11 January 2024
  • An '''entire function''' is a [[function]] <math>f:\mathbb{C}\to\mathbb{C}</math> that is [[holomorphic]] on the ful
    176 bytes (26 words) - 16:39, 28 March 2009
  • of the zeros of the [[Riemann zeta function]] and the distribution [[Riemann Hypothesis]], namely that the zeta function's nontrivial
    10 KB (1,729 words) - 18:52, 21 October 2023
  • ...two lines}\qquad\mathrm{(D)}\ \text{a circle}\qquad\mathrm{(E)}\ \text{the entire plane}</math> The function <math>f</math> has the property that for each real number <math>x</math> in
    15 KB (2,223 words) - 12:43, 28 December 2020
  • ...finite]]ly many) elements? You cannot list them all even if you spend your entire life writing! ...here is a bijective function <math>f:A\to B</math> meaning that there is a function <math>f</math> that maps all elements of <math>A</math> to all the elements
    11 KB (2,019 words) - 16:20, 7 July 2024
  • Let <math>f:(0,+\infty)\to\mathbb C</math> be a bounded function. Assume that ...y <math>T>0</math>, let <math>F_T(s) = \int_0^T f(t)e^{-st} dt</math>. The function
    6 KB (1,034 words) - 06:55, 12 August 2019
  • The function <math>f_{}^{}</math> has the property that, for each real number <math>x,\, ...rnal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to
    7 KB (1,141 words) - 06:37, 7 September 2018
  • ...+n^2</math>, it must divide <math>2n+1</math> if it is going to divide the entire [[expression]] <math>100+n^2+2n+1</math>. ...identity. Note for the LHS to be constant, <math>b</math> must be a linear function of <math>n,</math> so <math>b=px+q.</math> However, there cannot be a linea
    5 KB (795 words) - 21:27, 19 December 2024
  • So it is only necessary to find the length of the function at <math>3 \le x \le 4</math> and <math>3 \le y \le 4</math>: ...meter <math>4\sqrt{2}</math>. Now, we make use of the following fact for a function of two variables <math>x</math> and <math>y</math>: Suppose we have <math>f
    7 KB (1,225 words) - 18:56, 4 August 2021
  • ...t divides the trapezoid into two regions of equal area. Find the [[ceiling function|greatest integer]] that does not exceed <math>x^2/100</math>. ...a of the region including the shorter base must be half of the area of the entire trapezoid, so
    3 KB (433 words) - 18:42, 20 December 2021
  • A '''holomorphic function''' <math>f: \mathbb{C} \to \mathbb{C}</math> is a differentiable [[complex number|complex]] [[function]]. That is, just
    9 KB (1,537 words) - 20:04, 26 July 2017
  • ...owever, the function <math>f(x) = x^2</math> is ''not'' monotonic over the entire real line because it sometimes increases and sometimes decreases. ...grightarrow f(a) \leq f(b)</math> (resp. <math>f(a) \geq f(b)</math>. The function is ''strictly monotonic'' if, in addition, <math>a \neq b \Longrightarrow f
    1 KB (155 words) - 16:15, 22 August 2006
  • The function <math>f(x) = \sqrt{x(ax+b)}</math> has a [[codomain]] of all non-negative n ...ust be non-positive, <math>b</math> is non-negative, and the domain of the function occurs when <math>x(ax+b) > 0</math>, or
    9 KB (1,606 words) - 10:34, 10 July 2020
  • Form 1: The entire lower left <math>3\times2</math> rectangle is shaded, forcing the opposite ...uares in the row are shaded, then the row is represented by the generating function <math>ab+ac+ad+bc+bd+cd</math>, which we can write as <math>P(a,b,c,d)=(ab+
    13 KB (2,328 words) - 23:12, 28 November 2023
  • ...th> foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair <math>(a,b)</m A function <math>f</math> is defined by <math>f(z) = (4 + i) z^2 + \alpha z + \gamma</
    14 KB (2,199 words) - 12:43, 28 August 2020
  • ...thematician in the first or second room). Then look at <math>f</math> as a function into the [[field]] with two elements, <math>F_2</math>. Let <math>V</math> ...math>s on the diagonal is even. But since every vertex has odd degree, the entire diagonal of <math>A</math> consists of <math>1</math>s, so this says that t
    13 KB (2,414 words) - 13:37, 11 July 2016
  • ...nt of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half Let <math>f(x)</math> be a function with the two properties:
    15 KB (2,222 words) - 09:40, 11 August 2020
  • ...die so that he can remove all but <math>12</math> of the numbers from the entire sequence of rolls and (without changing the order of the sequence), be left Let <math>\phi(n)</math> denote <math>\textit{Euler's phi function}</math>, the number of integers <math>1\leq i\leq n</math> that are relativ
    71 KB (11,749 words) - 11:39, 20 November 2024

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