This property may also be cast in terms of convergent sequences in U. If such a number L exists, we usually denote it by ƒ'(w). More specifically, to say that ƒ: U → C is differentiable at an interior point w in U means the following: there exists some complex number L such that for every real number ε > 0 there exists a real number δ > 0 with the property that for all complex numbers z in U with 0 < |z - w| < δ, we have |/ - L| < ε. Holomorphic functions are central in the theory of complex functions. If ƒ is differentiable on an open set U, one also says that ƒ is holomorphic on U, or sometimes that ƒ is analytic on U.

If U is open, and if ƒ is differentiable on all of U, we say that ƒ is differentiable (on U). If the limitĮxists and is finite, we say that ƒ is differentiable at w, and we denote the value of this limit by ƒ'(w). Suppose ƒ: U → C is a complex-valued function defined on U, and suppose w is an interior point of U. Let C denote the set of complex numbers, and suppose U is some subset of C. As such, their properties are easily deduced from properties of the exponential function.Ĭomplex-valued functions defined on a subset of complex numbers may (some times) be differentiated, yes. The hyperbolic functions are usually defined in terms of the exponential function. It is important to know the inverse trig functions as they come in handy in many situations, like trig substitution in integral calculus. Or, using the term arcsin and not sin^-1 (though sin^-1 is more common) Like the square/square root example, if you have y=sin(x), which is y in terms of x, but you want to take that expression and find x in terms of y, then given: It is the same deal with sin and arcsin, which is conventionally written as sin^-1 x.Īrcsin is the inverse of sin, such that arcsin(sin(x)) = x, or sin(arcsin(x))=x. Now if you want to find out what x is in terms of y, then solve for x to get x=√y.Īs you know, the square operator and the square root operator are inverses of each other, that is, one 'undoes' the other: √(x²) = (√x)² = x (assuming we are only interested in the principal square root).

Trig derivative identities cheat sheet