Complex and real exponential #
In this file we prove continuity of Complex.exp
and Real.exp
. We also prove a few facts about
limits of Real.exp
at infinity.
Tags #
exp
The complex exponential function is uniformly continuous on left half planes.
Alias of ContinuousWithinAt.rexp
.
Alias of ContinuousAt.rexp
.
Alias of ContinuousOn.rexp
.
Alias of Continuous.rexp
.
The real exponential function tends to +∞
at +∞
.
The real exponential function tends to 0
at -∞
or, equivalently, exp(-x)
tends to 0
at +∞
Alias of Real.tendsto_exp_neg_atTop_nhds_zero
.
The real exponential function tends to 0
at -∞
or, equivalently, exp(-x)
tends to 0
at +∞
The real exponential function tends to 1
at 0
.
Alias of Real.tendsto_exp_nhds_zero_nhds_one
.
The real exponential function tends to 1
at 0
.
The function exp(x)/x^n
tends to +∞
at +∞
, for any natural number n
The function x^n * exp(-x)
tends to 0
at +∞
, for any natural number n
.
Alias of Real.tendsto_pow_mul_exp_neg_atTop_nhds_zero
.
The function x^n * exp(-x)
tends to 0
at +∞
, for any natural number n
.
Real.exp (f x)
is bounded away from zero along a filter if and only if this filter is bounded
from below under f
.
Real.exp (f x)
is bounded away from zero along a filter if and only if this filter is bounded
from below under f
.
Real.exp (f x)
is bounded away from zero and infinity along a filter l
if and only if
|f x|
is bounded from above along this filter.
Complex.abs (Complex.exp z) → ∞
as Complex.re z → ∞
.
Complex.exp z → 0
as Complex.re z → -∞
.