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Add functions and tests.

This commit is contained in:
William Rieger 2015-09-13 18:19:00 -04:00
parent 50d89519a2
commit c715405b30
1 changed files with 404 additions and 81 deletions
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485
src/complex.rs
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@ -98,74 +98,188 @@ impl<T: Clone + Float> Complex<T> {
Complex::new(*r * theta.cos(), *r * theta.sin())
}
/// Computes e^(self), where e is the base of the natural logarithm.
/// Computes `e^(self)`, where `e` is the base of the natural logarithm.
#[inline]
pub fn exp(&self) -> Complex<T> {
// formula: e^(a + bi) = e^a * (cos(b) + isin(b))
let exp = self.re.exp();
Complex::new(exp * self.im.cos(), exp * self.im.sin())
// formula: e^(a + bi) = e^a (cos(b) + i*sin(b))
Complex::new(self.im.clone().cos(), self.im.clone().sin()).scale(self.re.clone().exp())
}
/// Computes the sine of self.
/// Computes the principal value of natural logarithm of `self`.
///
/// This function has one branch cut:
///
/// * `(-∞, 0]`, continuous from above.
///
/// The branch satisfies `-π ≤ arg(ln(z)) ≤ π`.
#[inline]
pub fn ln(&self) -> Complex<T> {
// formula: ln(z) = ln|z| + i*arg(z)
Complex::new(self.norm().ln(), self.arg())
}
/// Computes the principal value of the square root of `self`.
///
/// This function has one branch cut:
///
/// * `(-∞, 0)`, continuous from above.
///
/// The branch satisfies `-π/2 ≤ arg(sqrt(z)) ≤ π/2`.
#[inline]
pub fn sqrt(&self) -> Complex<T> {
// formula: sqrt(r e^(it)) = sqrt(r) e^(it/2)
let two = T::one() + T::one();
let (r, theta) = self.to_polar();
Complex::from_polar(&(r.sqrt()), &(theta/two))
}
/// Computes the sine of `self`.
#[inline]
pub fn sin(&self) -> Complex<T> {
// formula: sin(z) = (e^(iz) - e^(-iz)) / 2i
//let one = One::one();
let i = Complex::new(Zero::zero(), One::one());
let two_i = i + i;
let e_iz = (self*i).exp();
let e_neg_iz = e_iz.inv();
(e_iz - e_neg_iz) / two_i
// formula: sin(a + bi) = sin(a)cosh(b) + i*cos(a)sinh(b)
Complex::new(self.re.clone().sin() * self.im.clone().cosh(), self.re.clone().cos() * self.im.clone().sinh())
}
/// Computes the cosine of self.
/// Computes the cosine of `self`.
#[inline]
pub fn cos(&self) -> Complex<T> {
// formula: cos(z) = (e^(iz) + e^(-iz)) / 2
let i = Complex::new(Zero::zero(), One::one());
let two = Complex::one() + Complex::one();
let e_iz = (self*i).exp();
let e_neg_iz = e_iz.inv();
(e_iz + e_neg_iz) / two
// formula: cos(a + bi) = cos(a)cosh(b) - i*sin(a)sinh(b)
Complex::new(self.re.clone().cos() * self.im.clone().cosh(), -self.re.clone().sin() * self.im.clone().sinh())
}
/// Computes the tangent of self.
/// Computes the tangent of `self`.
#[inline]
pub fn tan(&self) -> Complex<T> {
// formula: tan(z) = i (e^(-iz) - e^(iz)) / (e^(-iz) + e^(iz))
let i = Complex::new(Zero::zero(), One::one());
let e_iz = (self*i).exp();
let e_neg_iz = e_iz.inv();
i * (e_neg_iz - e_iz) / (e_neg_iz + e_iz)
// formula: tan(a + bi) = (sin(2a) + i*sinh(2b))/(cos(2a) + cosh(2b))
let (two_re, two_im) = (self.re.clone() + self.re.clone(), self.im.clone() + self.im.clone());
Complex::new(two_re.clone().sin(), two_im.clone().sinh()).unscale(two_re.cos() + two_im.cosh())
}
/// Computes the hyperbolic sine of self.
/// Computes the principal value of the inverse sine of `self`.
///
/// This function has two branch cuts:
///
/// * `(-∞, -1)`, continuous from above.
/// * `(1, ∞)`, continuous from below.
///
/// The branch satisfies `-π/2 ≤ Re(asin(z)) ≤ π/2`.
#[inline]
pub fn asin(&self) -> Complex<T> {
// formula: arcsin(z) = -i ln(sqrt(1-z^2) + iz)
let i = Complex::new(T::zero(), T::one());
-i*((Complex::one() - self*self).sqrt() + i*self).ln()
}
/// Computes the principal value of the inverse cosine of `self`.
///
/// This function has two branch cuts:
///
/// * `(-∞, -1)`, continuous from above.
/// * `(1, ∞)`, continuous from below.
///
/// The branch satisfies `0 ≤ Re(acos(z)) ≤ π`.
#[inline]
pub fn acos(&self) -> Complex<T> {
// formula: arccos(z) = -i ln(i sqrt(1-z^2) + z)
let i = Complex::new(T::zero(), T::one());
-i*(i*(Complex::one() - self*self).sqrt() + self).ln()
}
/// Computes the principal value of the inverse tangent of `self`.
///
/// This function has two branch cuts:
///
/// * `(-∞i, -i]`, continuous from the left.
/// * `[i, ∞i)`, continuous from the right.
///
/// The branch satisfies `-π/2 ≤ Re(atan(z)) ≤ π/2`.
#[inline]
pub fn atan(&self) -> Complex<T> {
// formula: arctan(z) = (ln(1+iz) - ln(1-iz))/(2i)
let i = Complex::new(T::zero(), T::one());
let one = Complex::one();
let two = one + one;
if *self == i {
return Complex::new(T::zero(), T::infinity());
}
else if *self == -i {
return Complex::new(T::zero(), -T::infinity());
}
((one + i * self).ln() - (one - i * self).ln()) / (two * i)
}
/// Computes the hyperbolic sine of `self`.
#[inline]
pub fn sinh(&self) -> Complex<T> {
// formula: sinh(z) = (e^(z) - e^(-z)) / 2
let two = Complex::one() + Complex::one();
let e_z = self.exp();
let e_neg_z = e_z.inv();
(e_z - e_neg_z) / two
// formula: sinh(a + bi) = sinh(a)cos(b) + i*cosh(a)sin(b)
Complex::new(self.re.clone().sinh() * self.im.clone().cos(), self.re.clone().cosh() * self.im.clone().sin())
}
/// Computes the hyperbolic cosine of self.
/// Computes the hyperbolic cosine of `self`.
#[inline]
pub fn cosh(&self) -> Complex<T> {
// formula: sinh(z) = (e^(z) + e^(-z)) / 2
let two = Complex::one() + Complex::one();
let e_z = self.exp();
let e_neg_z = e_z.inv();
(e_z + e_neg_z) / two
// formula: cosh(a + bi) = cosh(a)cos(b) + i*sinh(a)sin(b)
Complex::new(self.re.clone().cosh() * self.im.clone().cos(), self.re.clone().sinh() * self.im.clone().sin())
}
/// Computes the hyperbolic tangent of self.
/// Computes the hyperbolic tangent of `self`.
#[inline]
pub fn tanh(&self) -> Complex<T> {
// formula: tanh(z) = (e^(z) - e^(-z)) / (e^(z) + e^(-z))
let e_z = self.exp();
let e_neg_z = e_z.inv();
(e_z - e_neg_z) / (e_z + e_neg_z)
// formula: tanh(a + bi) = (sinh(2a) + i*sin(2b))/(cosh(2a) + cos(2b))
let (two_re, two_im) = (self.re.clone() + self.re.clone(), self.im.clone() + self.im.clone());
Complex::new(two_re.clone().sinh(), two_im.clone().sin()).unscale(two_re.cosh() + two_im.cos())
}
/// Computes the principal value of inverse hyperbolic sine of `self`.
///
/// This function has two branch cuts:
///
/// * `(-∞i, -i)`, continuous from the left.
/// * `(i, ∞i)`, continuous from the right.
///
/// The branch satisfies `-π/2 ≤ Im(asinh(z)) ≤ π/2`.
#[inline]
pub fn asinh(&self) -> Complex<T> {
// formula: arcsinh(z) = ln(z + sqrt(1+z^2))
let one = Complex::one();
(self + (one + self * self).sqrt()).ln()
}
/// Computes the principal value of inverse hyperbolic cosine of `self`.
///
/// This function has one branch cut:
///
/// * `(-∞, 1)`, continuous from above.
///
/// The branch satisfies `-π ≤ Im(acosh(z)) ≤ π` and `0 ≤ Re(acosh(z)) < ∞`.
#[inline]
pub fn acosh(&self) -> Complex<T> {
// formula: arccosh(z) = 2 ln(sqrt((z+1)/2) + sqrt((z-1)/2))
let one = Complex::one();
let two = one + one;
two * (((self + one)/two).sqrt() + ((self - one)/two).sqrt()).ln()
}
/// Computes the principal value of inverse hyperbolic tangent of `self`.
///
/// This function has two branch cuts:
///
/// * `(-∞, -1]`, continuous from above.
/// * `[1, ∞)`, continuous from below.
///
/// The branch satisfies `-π/2 ≤ Im(atanh(z)) ≤ π/2`.
#[inline]
pub fn atanh(&self) -> Complex<T> {
// formula: arctanh(z) = (ln(1+z) - ln(1-z))/2
let one = Complex::one();
let two = one + one;
if *self == one {
return Complex::new(T::infinity(), T::zero());
}
else if *self == -one {
return Complex::new(-T::infinity(), T::zero());
}
((one + self).ln() - (one - self).ln()) / two
}
}
@ -427,87 +541,296 @@ mod test {
for &c in all_consts.iter() { test(c); }
}
fn very_close(a: Complex64, b: Complex64) -> bool {
fn close(a: Complex64, b: Complex64) -> bool {
// returns true if a and b are reasonably close
(a-b).norm() < 1e-10
(a == b) || (a-b).norm() < 1e-10
}
#[test]
fn test_exp() {
assert_eq!(_1_0i.exp(), Complex::new(f64::consts::E, 0.0));
assert_eq!(_0_0i.exp(), _1_0i);
assert_eq!(_0_1i.exp(), Complex::new(1.0.cos(), 1.0.sin()));
assert!(very_close(_05_05i.exp()*_05_05i.exp(), _1_1i.exp()));
assert!(very_close(Complex::new(0.0, -f64::consts::PI).exp(), _1_0i.scale(-1.0)));
assert!(close(_1_0i.exp(), _1_0i.scale(f64::consts::E)));
assert!(close(_0_0i.exp(), _1_0i));
assert!(close(_0_1i.exp(), Complex::new(1.0.cos(), 1.0.sin())));
assert!(close(_05_05i.exp()*_05_05i.exp(), _1_1i.exp()));
assert!(close(_0_1i.scale(-f64::consts::PI).exp(), _1_0i.scale(-1.0)));
for &c in all_consts.iter() {
assert!(very_close(c.exp(), (c + Complex::new(0.0, f64::consts::PI*2.0)).exp()));
// e^conj(z) = conj(e^z)
assert!(close(c.conj().exp(), c.exp().conj()));
// e^(z + 2 pi i) = e^z
assert!(close(c.exp(), (c + _0_1i.scale(f64::consts::PI*2.0)).exp()));
}
}
#[test]
fn test_ln() {
assert!(close(_1_0i.ln(), _0_0i));
assert!(close(_0_1i.ln(), _0_1i.scale(f64::consts::PI/2.0)));
assert!(close(_0_0i.ln(), Complex::new(f64::neg_infinity(), 0.0)));
assert!(close((_neg1_1i * _05_05i).ln(), _neg1_1i.ln() + _05_05i.ln()));
for &c in all_consts.iter() {
// ln(conj(z() = conj(ln(z))
assert!(close(c.conj().ln(), c.ln().conj()));
// for this branch, -pi <= arg(ln(z)) <= pi
assert!(-f64::consts::PI <= c.ln().arg() && c.ln().arg() <= f64::consts::PI);
}
}
#[test]
fn test_sqrt() {
assert!(close(_0_0i.sqrt(), _0_0i));
assert!(close(_1_0i.sqrt(), _1_0i));
assert!(close(Complex::new(-1.0, 0.0).sqrt(), _0_1i));
assert!(close(Complex::new(-1.0, -0.0).sqrt(), _0_1i.scale(-1.0)));
assert!(close(_0_1i.sqrt(), _05_05i.scale(2.0.sqrt())));
for &c in all_consts.iter() {
// sqrt(conj(z() = conj(sqrt(z))
assert!(close(c.conj().sqrt(), c.sqrt().conj()));
// for this branch, -pi/2 <= arg(sqrt(z)) <= pi/2
assert!(-f64::consts::PI/2.0 <= c.sqrt().arg() && c.sqrt().arg() <= f64::consts::PI/2.0);
// sqrt(z) * sqrt(z) = z
assert!(close(c.sqrt()*c.sqrt(), c));
}
}
#[test]
fn test_sin() {
assert_eq!(_0_0i.sin(), _0_0i);
assert!(very_close(_1_0i.scale(f64::consts::PI*2.0).sin(), _0_0i));
assert_eq!(_0_1i.sin(), _0_1i.scale(1.0.sinh()));
assert!(close(_0_0i.sin(), _0_0i));
assert!(close(_1_0i.scale(f64::consts::PI*2.0).sin(), _0_0i));
assert!(close(_0_1i.sin(), _0_1i.scale(1.0.sinh())));
for &c in all_consts.iter() {
assert!(very_close(c.conj().sin(), c.sin().conj()));
assert!(very_close(c.scale(-1.0).sin(), c.sin().scale(-1.0)));
// sin(conj(z)) = conj(sin(z))
assert!(close(c.conj().sin(), c.sin().conj()));
// sin(-z) = -sin(z)
assert!(close(c.scale(-1.0).sin(), c.sin().scale(-1.0)));
}
}
#[test]
fn test_cos() {
assert_eq!(_0_0i.cos(), _1_0i);
assert!(very_close(_1_0i.scale(f64::consts::PI*2.0).cos(), _1_0i));
assert_eq!(_0_1i.cos(), _1_0i.scale(1.0.cosh()));
assert!(close(_0_0i.cos(), _1_0i));
assert!(close(_1_0i.scale(f64::consts::PI*2.0).cos(), _1_0i));
assert!(close(_0_1i.cos(), _1_0i.scale(1.0.cosh())));
for &c in all_consts.iter() {
assert!(very_close(c.conj().cos(), c.cos().conj()));
assert!(very_close(c.scale(-1.0).cos(), c.cos()));
// cos(conj(z)) = conj(cos(z))
assert!(close(c.conj().cos(), c.cos().conj()));
// cos(-z) = cos(z)
assert!(close(c.scale(-1.0).cos(), c.cos()));
}
}
#[test]
fn test_tan() {
assert_eq!(_0_0i.tan(), _0_0i);
assert!(very_close(_1_0i.scale(f64::consts::PI).tan(), _0_0i));
assert!(close(_0_0i.tan(), _0_0i));
assert!(close(_1_0i.scale(f64::consts::PI/4.0).tan(), _1_0i));
assert!(close(_1_0i.scale(f64::consts::PI).tan(), _0_0i));
for &c in all_consts.iter() {
assert!(very_close(c.conj().tan(), c.tan().conj()));
assert!(very_close(c.scale(-1.0).tan(), c.tan().scale(-1.0)));
assert!(very_close(c.tan(), c.sin()/c.cos()));
// tan(conj(z)) = conj(tan(z))
assert!(close(c.conj().tan(), c.tan().conj()));
// tan(-z) = -tan(z)
assert!(close(c.scale(-1.0).tan(), c.tan().scale(-1.0)));
}
}
#[test]
fn test_asin() {
assert!(close(_0_0i.asin(), _0_0i));
assert!(close(_1_0i.asin(), _1_0i.scale(f64::consts::PI/2.0)));
assert!(close(_1_0i.scale(-1.0).asin(), _1_0i.scale(-f64::consts::PI/2.0)));
assert!(close(_0_1i.asin(), _0_1i.scale((1.0 + 2.0.sqrt()).ln())));
for &c in all_consts.iter() {
// asin(conj(z)) = conj(asin(z))
assert!(close(c.conj().asin(), c.asin().conj()));
// asin(-z) = -asin(z)
assert!(close(c.scale(-1.0).asin(), c.asin().scale(-1.0)));
// for this branch, -pi/2 <= asin(z).re <= pi/2
assert!(-f64::consts::PI/2.0 <= c.asin().re && c.asin().re <= f64::consts::PI/2.0);
}
}
#[test]
fn test_acos() {
assert!(close(_0_0i.acos(), _1_0i.scale(f64::consts::PI/2.0)));
assert!(close(_1_0i.acos(), _0_0i));
assert!(close(_1_0i.scale(-1.0).acos(), _1_0i.scale(f64::consts::PI)));
assert!(close(_0_1i.acos(), Complex::new(f64::consts::PI/2.0, (2.0.sqrt() - 1.0).ln())));
for &c in all_consts.iter() {
// acos(conj(z)) = conj(acos(z))
assert!(close(c.conj().acos(), c.acos().conj()));
// for this branch, 0 <= acos(z).re <= pi
assert!(0.0 <= c.acos().re && c.acos().re <= f64::consts::PI);
}
}
#[test]
fn test_atan() {
assert!(close(_0_0i.atan(), _0_0i));
assert!(close(_1_0i.atan(), _1_0i.scale(f64::consts::PI/4.0)));
assert!(close(_1_0i.scale(-1.0).atan(), _1_0i.scale(-f64::consts::PI/4.0)));
assert!(close(_0_1i.atan(), Complex::new(0.0, f64::infinity())));
for &c in all_consts.iter() {
// atan(conj(z)) = conj(atan(z))
assert!(close(c.conj().atan(), c.atan().conj()));
// atan(-z) = -atan(z)
assert!(close(c.scale(-1.0).atan(), c.atan().scale(-1.0)));
// for this branch, -pi/2 <= atan(z).re <= pi/2
assert!(-f64::consts::PI/2.0 <= c.atan().re && c.atan().re <= f64::consts::PI/2.0);
}
}
#[test]
fn test_sinh() {
assert_eq!(_0_0i.sinh(), _0_0i);
assert_eq!(_1_0i.sinh(), _1_0i.scale((f64::consts::E - 1.0/f64::consts::E)/2.0));
assert_eq!(_0_1i.sinh(), _0_1i.scale(1.0.sin()));
assert!(close(_0_0i.sinh(), _0_0i));
assert!(close(_1_0i.sinh(), _1_0i.scale((f64::consts::E - 1.0/f64::consts::E)/2.0)));
assert!(close(_0_1i.sinh(), _0_1i.scale(1.0.sin())));
for &c in all_consts.iter() {
assert!(very_close(c.conj().sinh(), c.sinh().conj()));
assert!(very_close(c.scale(-1.0).sinh(), c.sinh().scale(-1.0)));
// sinh(conj(z)) = conj(sinh(z))
assert!(close(c.conj().sinh(), c.sinh().conj()));
// sinh(-z) = -sinh(z)
assert!(close(c.scale(-1.0).sinh(), c.sinh().scale(-1.0)));
}
}
#[test]
fn test_cosh() {
assert_eq!(_0_0i.cosh(), _1_0i);
assert_eq!(_1_0i.cosh(), _1_0i.scale((f64::consts::E + 1.0/f64::consts::E)/2.0));
assert_eq!(_0_1i.cosh(), _1_0i.scale(1.0.cos()));
assert!(close(_0_0i.cosh(), _1_0i));
assert!(close(_1_0i.cosh(), _1_0i.scale((f64::consts::E + 1.0/f64::consts::E)/2.0)));
assert!(close(_0_1i.cosh(), _1_0i.scale(1.0.cos())));
for &c in all_consts.iter() {
assert!(very_close(c.conj().cosh(), c.cosh().conj()));
assert!(very_close(c.scale(-1.0).cosh(), c.cosh()));
// cosh(conj(z)) = conj(cosh(z))
assert!(close(c.conj().cosh(), c.cosh().conj()));
// cosh(-z) = cosh(z)
assert!(close(c.scale(-1.0).cosh(), c.cosh()));
}
}
#[test]
fn test_tanh() {
assert_eq!(_0_0i.tanh(), _0_0i);
assert!(very_close(_1_0i.tanh(), _1_0i.scale((f64::consts::E.powi(2) - 1.0)/(f64::consts::E.powi(2) + 1.0))));
assert!(very_close(_0_1i.tanh(), _0_1i.scale(1.0.tan())));
assert!(close(_0_0i.tanh(), _0_0i));
assert!(close(_1_0i.tanh(), _1_0i.scale((f64::consts::E.powi(2) - 1.0)/(f64::consts::E.powi(2) + 1.0))));
assert!(close(_0_1i.tanh(), _0_1i.scale(1.0.tan())));
for &c in all_consts.iter() {
assert!(very_close(c.conj().tanh(), c.conj().tanh()));
assert!(very_close(c.scale(-1.0).tanh(), c.tanh().scale(-1.0)));
assert!(very_close(c.tanh(), c.sinh()/c.cosh()));
// tanh(conj(z)) = conj(tanh(z))
assert!(close(c.conj().tanh(), c.conj().tanh()));
// tanh(-z) = -tanh(z)
assert!(close(c.scale(-1.0).tanh(), c.tanh().scale(-1.0)));
}
}
#[test]
fn test_asinh() {
assert!(close(_0_0i.asinh(), _0_0i));
assert!(close(_1_0i.asinh(), _1_0i.scale(1.0 + 2.0.sqrt()).ln()));
assert!(close(_0_1i.asinh(), _0_1i.scale(f64::consts::PI/2.0)));
assert!(close(_0_1i.asinh().scale(-1.0), _0_1i.scale(-f64::consts::PI/2.0)));
for &c in all_consts.iter() {
// asinh(conj(z)) = conj(asinh(z))
assert!(close(c.conj().asinh(), c.conj().asinh()));
// asinh(-z) = -asinh(z)
assert!(close(c.scale(-1.0).asinh(), c.asinh().scale(-1.0)));
// for this branch, -pi/2 <= asinh(z).im <= pi/2
assert!(-f64::consts::PI/2.0 <= c.asinh().im && c.asinh().im <= f64::consts::PI/2.0);
}
}
#[test]
fn test_acosh() {
assert!(close(_0_0i.acosh(), _0_1i.scale(f64::consts::PI/2.0)));
assert!(close(_1_0i.acosh(), _0_0i));
assert!(close(_1_0i.scale(-1.0).acosh(), _0_1i.scale(f64::consts::PI)));
for &c in all_consts.iter() {
// acosh(conj(z)) = conj(acosh(z))
assert!(close(c.conj().acosh(), c.conj().acosh()));
// for this branch, -pi <= acosh(z).im <= pi and 0 <= acosh(z).re
assert!(-f64::consts::PI <= c.acosh().im && c.acosh().im <= f64::consts::PI && 0.0 <= c.cosh().re);
}
}
#[test]
fn test_atanh() {
assert!(close(_0_0i.atanh(), _0_0i));
assert!(close(_0_1i.atanh(), _0_1i.scale(f64::consts::PI/4.0)));
assert!(close(_1_0i.atanh(), Complex::new(f64::infinity(), 0.0)));
for &c in all_consts.iter() {
// atanh(conj(z)) = conj(atanh(z))
assert!(close(c.conj().atanh(), c.conj().atanh()));
// atanh(-z) = -atanh(z)
assert!(close(c.scale(-1.0).atanh(), c.atanh().scale(-1.0)));
// for this branch, -pi/2 <= atanh(z).im <= pi/2
assert!(-f64::consts::PI/2.0 <= c.atanh().im && c.atanh().im <= f64::consts::PI/2.0);
}
}
#[test]
fn test_exp_ln() {
for &c in all_consts.iter() {
// e^ln(z) = z
assert!(close(c.ln().exp(), c));
}
}
#[test]
fn test_trig_to_hyperbolic() {
for &c in all_consts.iter() {
// sin(iz) = i sinh(z)
assert!(close((_0_1i * c).sin(), _0_1i * c.sinh()));
// cos(iz) = cosh(z)
assert!(close((_0_1i * c).cos(), c.cosh()));
// tan(iz) = i tanh(z)
assert!(close((_0_1i * c).tan(), _0_1i * c.tanh()));
}
}
#[test]
fn test_trig_identities() {
for &c in all_consts.iter() {
// tan(z) = sin(z)/cos(z)
assert!(close(c.tan(), c.sin()/c.cos()));
// sin(z)^2 + cos(z)^2 = 1
assert!(close(c.sin()*c.sin() + c.cos()*c.cos(), _1_0i));
// sin(asin(z)) = z
assert!(close(c.asin().sin(), c));
// cos(acos(z)) = z
assert!(close(c.acos().cos(), c));
// tan(atan(z)) = z
// i and -i are branch points
if c != _0_1i && c != _0_1i.scale(-1.0) {
assert!(close(c.atan().tan(), c));
}
// sin(z) = (e^(iz) - e^(-iz))/(2i)
assert!(close(((_0_1i*c).exp() - (_0_1i*c).exp().inv())/_0_1i.scale(2.0), c.sin()));
// cos(z) = (e^(iz) + e^(-iz))/2
assert!(close(((_0_1i*c).exp() + (_0_1i*c).exp().inv()).unscale(2.0), c.cos()));
// tan(z) = i (1 - e^(2iz))/(1 + e^(2iz))
assert!(close(_0_1i * (_1_0i - (_0_1i*c).scale(2.0).exp())/(_1_0i + (_0_1i*c).scale(2.0).exp()), c.tan()));
}
}
#[test]
fn test_hyperbolic_identites() {
for &c in all_consts.iter() {
// tanh(z) = sinh(z)/cosh(z)
assert!(close(c.tanh(), c.sinh()/c.cosh()));
// cosh(z)^2 - sinh(z)^2 = 1
assert!(close(c.cosh()*c.cosh() - c.sinh()*c.sinh(), _1_0i));
// sinh(asinh(z)) = z
assert!(close(c.asinh().sinh(), c));
// cosh(acosh(z)) = z
assert!(close(c.acosh().cosh(), c));
// tanh(atanh(z)) = z
// 1 and -1 are branch points
if c != _1_0i && c != _1_0i.scale(-1.0) {
assert!(close(c.atanh().tanh(), c));
}
// sinh(z) = (e^z - e^(-z))/2
assert!(close((c.exp() - c.exp().inv()).unscale(2.0), c.sinh()));
// cosh(z) = (e^z + e^(-z))/2
assert!(close((c.exp() + c.exp().inv()).unscale(2.0), c.cosh()));
// tanh(z) = ( e^(2z) - 1)/(e^(2z) + 1)
assert!(close((c.scale(2.0).exp() - _1_0i)/(c.scale(2.0).exp() + _1_0i), c.tanh()));
}
}