mathlib documentation

ring_theory.power_series.well_known

Definition of well-known power series #

In this file we define the following power series:

def power_series.inv_units_sub {R : Type u_1} [ring R] (u : units R) :

The power series for 1 / (u - x).

Equations
@[simp]
theorem power_series.coeff_inv_units_sub {R : Type u_1} [ring R] (u : units R) (n : ) :
theorem power_series.map_inv_units_sub {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R →+* S) (u : units R) :
def power_series.exp (A : Type u_1) [ring A] [algebra A] :

Power series for the exponential function at zero.

Equations
def power_series.sin (A : Type u_1) [ring A] [algebra A] :

Power series for the sine function at zero.

Equations
def power_series.cos (A : Type u_1) [ring A] [algebra A] :

Power series for the cosine function at zero.

Equations
@[simp]
theorem power_series.coeff_exp {A : Type u_1} [ring A] [algebra A] (n : ) :
@[simp]
theorem power_series.map_exp {A : Type u_1} {A' : Type u_2} [ring A] [ring A'] [algebra A] [algebra A'] (f : A →+* A') :
@[simp]
theorem power_series.map_sin {A : Type u_1} {A' : Type u_2} [ring A] [ring A'] [algebra A] [algebra A'] (f : A →+* A') :
@[simp]
theorem power_series.map_cos {A : Type u_1} {A' : Type u_2} [ring A] [ring A'] [algebra A] [algebra A'] (f : A →+* A') :

Shows that $e^{aX} * e^{bX} = e^{(a + b)X}$

Shows that $e^{x} * e^{-x} = 1$

Shows that $(e^{X})^k = e^{kX}$.

theorem power_series.exp_pow_sum {A : Type u_1} [comm_ring A] [algebra A] (n : ) :
∑ (k : ) in finset.range n, power_series.exp A ^ k = power_series.mk (λ (p : ), ∑ (k : ) in finset.range n, (k ^ p) * (algebra_map A) (p!)⁻¹)

Shows that $\sum_{k = 0}^{n - 1} (e^{X})^k = \sum_{p = 0}^{\infty} \sum_{k = 0}^{n - 1} \frac{k^p}{p!}X^p$.