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  2. Matrix exponential - Wikipedia

    en.wikipedia.org/wiki/Matrix_exponential

    e aX e bX = e (a + b)X; e X eX = I; Using the above results, we can easily verify the following claims. If X is symmetric then e X is also symmetric, and if X is skew-symmetric then e X is orthogonal. If X is Hermitian then e X is also Hermitian, and if X is skew-Hermitian then e X is unitary.

  3. Linear differential equation - Wikipedia

    en.wikipedia.org/wiki/Linear_differential_equation

    The study of these differential equations with constant coefficients dates back to Leonhard Euler, who introduced the exponential function e x, which is the unique solution of the equation f′ = f such that f(0) = 1. It follows that the n th derivative of e cx is c n e cx, and this allows solving homogeneous linear differential equations ...

  4. Exponential function - Wikipedia

    en.wikipedia.org/wiki/Exponential_function

    For instance, e x can be defined as (+). Or e x can be defined as f x (1), where f x : R → B is the solution to the differential equation ⁠ df x / dt ⁠ (t) = x f x (t), with initial condition f x (0) = 1; it follows that f x (t) = e tx for every t in R.

  5. List of integrals of exponential functions - Wikipedia

    en.wikipedia.org/wiki/List_of_integrals_of...

    The last expression is the logarithmic mean. = (⁡ >) = (>) (the Gaussian integral) = (>) = (, >) (+) = (>)(+ +) = (>)= (>) (see Integral of a Gaussian function

  6. Differential equation - Wikipedia

    en.wikipedia.org/wiki/Differential_equation

    In all these cases, y is an unknown function of x (or of x 1 and x 2), and f is a given function. He solves these examples and others using infinite series and discusses the non-uniqueness of solutions. Jacob Bernoulli proposed the Bernoulli differential equation in 1695. [3] This is an ordinary differential equation of the form

  7. Characterizations of the exponential function - Wikipedia

    en.wikipedia.org/wiki/Characterizations_of_the...

    Define () = to be the unique solution to the differential equation with initial value: ′ =, =, where ′ = denotes the derivative of y. Functional equation. The exponential function e x {\displaystyle e^{x}} is the unique function f with the multiplicative property f ( x + y ) = f ( x ) f ( y ) {\displaystyle f(x+y)=f(x)f(y)} for all x , y ...

  8. Exponential integral - Wikipedia

    en.wikipedia.org/wiki/Exponential_integral

    with the derivative evaluated at = Another connexion with the confluent hypergeometric functions is that E 1 is an exponential times the function U(1,1,z): = (,,) The exponential integral is closely related to the logarithmic integral function li(x) by the formula

  9. Differential calculus - Wikipedia

    en.wikipedia.org/wiki/Differential_calculus

    When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once.

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