# test_trig_series.py3  power series for a few functions
# just a few of the many possible series and equivalent expressions
# sin(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! ... reduce 0 <= x <= 2pi
# cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! ... reduce -pi <= x <= pi
# tan(x) = sin(x)/cos(x)
# csc(x) = 1/sin(x)          cosecant
# sec(x) = 1/cos(x)          secant
# cot(x) = 1/tan(x)          cotangent
# sinh(x) = (e^x - e^-x)/2 = x + x^3/3! + x^5/5! + x^7/7! ...
# cosh(x) = (e^x + e^-x)/2 = 1 + x^2/2! + x^4/4! + x^6/6! ...
# asin(x) = x + x^3/6 + 3*x^5/40 + 15*x^7/336  ...
# acos(x) = pi/2 - (x + x^3/6 + 3*x^4/40 + 15*x^7/336 ...)
# atan(x) = x - x^3/3 + x^5/5 - x^7/7 + x^9/9  ... 
# pi = 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13  ...)
# tanh(x) = sinh(x)/cosh(x)
# exp(x) = 1 + x + x^2/2! + x^3/3! + x4/4! ...
# e = 1 + 1 + 1/2! + 1/3! + 1/4!  ...
# log(x) = (x-1)/x + ((x-1)/x))^2/2 + ((x-1/x))^3/3 ...
import math

def fac(n) :
  fct = 1
  for i in range(2,n+1) :
    fct = fct * i

  # end i
  return float(fct)

# end fac

def pwr(x,n) :
  pwr = 1.0
  for i in range(n) :
    pwr = pwr * x

  # end i
  return pwr

# end pwr

print("test_trig_series.py3 running")
x = 0.5 
# sin(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! ... reduce 0 <= x <= 2pi
sinx = x - pwr(x,3)/fac(3) +  pwr(x,5)/fac(5) -  pwr(x,7)/fac(7) +  pwr(x,9)/fac(9)
print("sin(0.5)=",math.sin(0.5)," ser=",sinx)
print(" ")
# cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! ... reduce -pi <= x <= pi
cosx = 1.0 - pwr(x,2)/fac(2) + pwr(x,4)/fac(4) - pwr(x,6)/fac(6) + pwr(x,8)/fac(8)
print("cos(0.5)=",math.cos(0.5)," ser=",cosx)
print(" ")
# tan(x) = sin(x)/cos(x)
tanx = sinx/cosx
print("tan(0.5)=",math.tan(0.5)," ser=",tanx)
print(" ")
# sinh(x) = x + x^3/3! + x^5/5! + x^7/7! + x^9/9! ... reduce 0 <= x <= 2pi
sinhx = x + pwr(x,3)/fac(3) +  pwr(x,5)/fac(5) + pwr(x,7)/fac(7) + pwr(x,9)/fac(9)
print("sinh(0.5)=",math.sinh(0.5)," ser=",sinhx)
print(" ")
# cosh(x) = 1 + x^2/2! + x^4/4! + x^6/6! ...
coshx = 1.0 + pwr(x,2)/fac(2) +  pwr(x,4)/fac(4) + pwr(x,6)/fac(6) + pwr(x,8)/fac(8)
print("cosh(0.5)=",math.cosh(0.5)," ser=",coshx)
print(" ")
# asin(x) = x + x^3/6 + 3*x^5/40 + 15*x^7/336  ...
asinx = x + pwr(x,3)/6.0 + 3.0*pwr(x,5)/40.0 + 15.0*pwr(x,7)/336.0
print("asin(0.5)=",math.asin(0.5)," ser=",asinx)
print(" ")
# acos(x) = pi/2 - (x + x^3/6 + 3*x^5/40 + 15*x^7/336 ...)
acosx = math.pi/2.0 - (x + pwr(x,3)/6.0 + 3.0*pwr(x,5)/40.0 + 15.0*pwr(x,7)/336.0)
print("acos(0.5)=",math.acos(0.5)," ser=",acosx)
print(" ")
# atan(x) = x - x^3/3 + x^5/5 - x^7/7 + x^9/9  ... 
atanx = x - pwr(x,3)/3.0 + pwr(x,5)/5.0 - pwr(x,7)/7.0 + pwr(x,9)/9.0  
print("atan(0.5)=",math.atan(0.5)," ser=",atanx)
print(" ")
# pi = 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15  ...) # at least 51
pi4ser = 1.0 - 1.0/3.0 + 1.0/5.0 - 1.0/7.0 + 1.0/9.0 - 1.0/11.0 + 1.0/13.0 - 1.0/15.0 + 1.0/17.0 - 1.0/19.0 + 1.0/21.0 - 1.0/23.0 + 1.0/25.0 - 1.0/27.0 + 1.0/29.0 - 1.0/31.0 + 1.0/33.0 - 1.0/35.0 + 1.0/37.0 - 1.0/39.0 + 1.0/41.0  - 1.0/43.0 + 1.0/45.0  - 1.0/47.0 + 1.0/49.0  - 1.0/51.0 + 1.0/53.0 - 1.0/55.0 
piser = 4.0 * pi4ser
print("math.pi/4=",(math.pi/4.0)," ser=",pi4ser)
print(" ")
print("math.pi=",(math.pi)," ser=",piser)
print(" ")
# tanh(x) = sinh(x)/cosh(x)
tanhx = sinhx/coshx
print("math.tanh(x)=",math.tanh(x)," ser=",tanhx)
print(" ")
# exp(x) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5!  ...
expx = 1.0 + x + pwr(x,2)/fac(2) + pwr(x,3)/fac(3) + pwr(x,4)/fac(4) + pwr(x,5)/fac(5)
print("math.exp(0.5)=",math.exp(0.5)," ser=",expx)
print(" ")
# e = 1 + 1 + 1/2! + 1/3! + 1/4! * 1/5!  ...
eser = 1.0 + 1.0 + 1.0/fac(2) + 1.0/fac(3) + 1.0/fac(4) + 1.0/fac(5)
print("math.e=",math.e," ser=",eser)
print(" ")
# log(x) = (x-1)/x + ((x-1)/x))^2/2 + ((x-1/x))^3/3 + ((x-1/x))^4/4 ...
xt = (x-1)/x 
logx = xt + pwr(xt,2)/2.0 + pwr(xt,3)/3.0 + pwr(xt,4)/4.0
print("math.log(0.5)=",math.log(0.5)," ser=",logx)
print(" ")
print("test_trig_series.py3 finished")