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  2. Hyperbolic functions - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_functions

    The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers. The graph of the function a cosh( x / a ) is the catenary , the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.

  3. Hyperbolic partial differential equation - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_partial...

    In mathematics, a hyperbolic partial differential equation of order is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first derivatives. [ citation needed ] More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface .

  4. Differentiation rules - Wikipedia

    en.wikipedia.org/wiki/Differentiation_rules

    Differentiation of integrals – Problem in mathematics; Differentiation under the integral sign – Differentiation under the integral sign formula; Hyperbolic functions – Collective name of 6 mathematical functions; Inverse hyperbolic functions – Mathematical functions; Inverse trigonometric functions – Inverse functions of sin, cos ...

  5. Lax–Friedrichs method - Wikipedia

    en.wikipedia.org/wiki/Lax–Friedrichs_method

    A nonlinear hyperbolic conservation law is defined through a flux function : + (()) = In the case of f ( u ) = a u {\displaystyle f(u)=au} , we end up with a scalar linear problem. Note that in general, u {\displaystyle u} is a vector with m {\displaystyle m} equations in it.

  6. Bessel function - Wikipedia

    en.wikipedia.org/wiki/Bessel_function

    Bessel functions describe the radial part of vibrations of a circular membrane.. Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, [1] are canonical solutions y(x) of Bessel's differential equation + + = for an arbitrary complex number, which represents the order of the Bessel function.

  7. Dirac delta function - Wikipedia

    en.wikipedia.org/wiki/Dirac_delta_function

    The problems with a classical ... The first derivative of the delta function is the distributional ... the equations involved are hyperbolic and so may have ...

  8. Schwarz lemma - Wikipedia

    en.wikipedia.org/wiki/Schwarz_lemma

    The Schwarz–Ahlfors–Pick theorem provides an analogous theorem for hyperbolic manifolds. De Branges' theorem , formerly known as the Bieberbach Conjecture, is an important extension of the lemma, giving restrictions on the higher derivatives of f {\displaystyle f} at 0 {\displaystyle 0} in case f {\displaystyle f} is injective ; that is ...

  9. Method of characteristics - Wikipedia

    en.wikipedia.org/wiki/Method_of_characteristics

    Although this is defined using a particular coordinate system, the transformation law relating the ξ i and the x i ensures that σ P is a well-defined function on the cotangent bundle. The function σ P is homogeneous of degree k in the ξ variable. The zeros of σ P, away from the zero section of T ∗ X, are the characteristics of P.