Edexcel Further Mathematics: Cp2

Vocabulary flashcards covering key algebraic, complex number, series, and hyperbolic function concepts from the Core Pure Mathematics lecture notes.

Euler's Relation

The formula e^{i\theta} = \cos \theta + i \sin \theta.

Modulus-Argument Form

Representing a complex number as z = r(\cos \theta + i \sin \theta), where r is the distance from the origin and \theta is the angle from the positive real axis.

De Moivre's Theorem

A theorem stating that for any integer n, [r(\cos \theta + i \sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta).

Exponential Form

Writing a complex number in the form z = re^{i\theta}.

nth Roots of Unity

The solutions to the equation z^n = 1, which form the vertices of a regular n-gon on an Argand diagram.

Method of Differences

A technique to find the sum of a finite series by expressing the general term as a difference of two related terms, leading to a telescoping sum.

Maclaurin Series

An expansion of a function as an infinite sum of terms calculated from the values of its derivatives at zero.

Convergent Integral

An improper integral that tends to a finite limit as its bounds approach infinity or a point of discontinuity.

Hyperbolic Sine (sinh x)

Defined as \frac{e^x - e^{-x}}{2}.

Hyperbolic Cosine (cosh x)

Defined as \frac{e^x + e^{-x}}{2}.

Osborn's Rule

A mnemonic for converting trigonometric identities to hyperbolic identities by replacing \cos with \cosh and \sin with \sinh, but negating products of two sines.

Fundamental Hyperbolic Identity

The identity \cosh^{2} x - \sinh^{2} x = 1.

Inverse Hyperbolic Sine (arsinh x)

The logarithmic form is \ln(x + \sqrt{x^2 + 1}) for all real x.

Inverse Hyperbolic Cosine (arcosh x)

The logarithmic form is \ln(x + \sqrt{x^2 - 1}) for x \ge 1.

Differentiation of Hyperbolic Functions

\frac{d}{dx}(\sinh x) = \cosh x and \frac{d}{dx}(\cosh x) = \sinh x.

Taylor Series

A power series centered at x = a given by f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n.

Complex Conjugate Root Theorem

States that if a polynomial has real coefficients, then any complex roots must occur in conjugate pairs (z and z^*).