# 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  ...)
# 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 expser(x,n) : # n terms to  x^n / n!
  # print("expser(",x,",",n,") running")
  sum = 1.0 + x  # 0,1 i
  # print("sum=",sum)
  for i in range(2,n+1) :
    # print("i=",i)
    nfac = factorial(i)
    # print("nfac=",nfac)
    npwr = power(x,i)
    # print("npwr=",npwr)
    sum = sum + npwr/nfac

  # end i
  # print("expcer sum=",sum)
  # print(" ")
  return sum

# end expser
  
def sinser(x,n) : # n terms to  (-1)^n * x^(2n+1) / (2n+1)!
  # print("sinser(",x,",",n,") running")
  sum = 0.0
  sgn = 1.0
  for i in range(n) :
    j = 2*i+1
    # print("sincer i=",i," j=",j)
    nfac = factorial(j)
    # print("nfac=",nfac)
    npwr = power(x,j)
    # print("npwr=",npwr)
    sum = sum + sgn*npwr/nfac
    sgn = -sgn
    # print("sincer sum=",sum)

  # end i
  # print("sincer sum=",sum)
  # print(" ")
  return sum

# end sinser

def cosser(x,n) : # n terms to  (-1)^(n+1) * x^(2n) / (2n)!
  # print("cosser(",x,",",n,") running")
  sum = 0.0
  sgn = 1.0
  for i in range(n) :
    j = 2*i
    # print("cosser i=",i," j=",j)
    nfac = factorial(j)
    # print("nfac=",nfac)
    npwr = power(x,j)
    # print("npwr=",npwr)
    sum = sum + sgn*npwr/nfac
    sgn = -sgn
    # print("coscer sum=",sum)

  # end i
  # print("coscer sum=",sum)
  # print(" ")
  return sum

# end cosser

def sinhser(x,n) : # n terms to  x^(2n+1) / (2n+1)!
  # print("sinhser(",x,",",n,") running")
  sum = 0.0
  for i in range(n) :
    j = 2*i+1
    # print("j=",j)
    nfac = factorial(j)
    # print("nfac=",nfac)
    npwr = power(x,j)
    # print("npwr=",npwr)
    sum = sum + npwr/nfac
    # print("sinhcer sum=",sum)

  # end i
  # print("sinhcer sum=",sum)
  # print(" ")
  return sum

# end sinhser

def coshser(x,n) : # n terms to  x^(2n) / (2n)!
  # print("coshser(",x,",",n,") running")
  sum = 0.0
  for i in range(n) :
    j = 2*i
    # print("j=",j)
    nfac = factorial(j)
    # print("nfac=",nfac)
    npwr = power(x,j)
    # print("npwr=",npwr)
    sum = sum + npwr/nfac
    # print("coshcer sum=",sum)

  # end i
  # print("coshcer sum=",sum)
  # print(" ")
  return sum

# end coshser

def logser(x,n) : # n terms to  (-1)^n * y^n / n!
  # print("logser(",x,",",n,") running")
  sum = 0.0
  sgn = 1.0
  y = x-1.0
  for i in range(1,n) :
    # print("i=",i," sgn=",sgn)
    nfac = factorial(i)
    # print("nfac=",nfac)
    npwr = power(y,i)
    # print("npwr=",npwr)
    sum = sum + sgn*npwr/nfac
    sgn = -sgn
    # print("logcer sum=",sum)

  # end i
  # print("logcer sum=",sum)
  # print(" ")
  return sum

# end logser

def logp1ser(x,n) : # n terms to  (-1)^n * y^n / n!
  # print("logp1ser(",x,",",n,") running")
  sum = 0.0
  sgn = 1.0
  for i in range(1,n) :
    # print("i=",i," sgn=",sgn)
    nfac = factorial(i)
    # print("nfac=",nfac)
    npwr = power(x,i)
    # print("npwr=",npwr)
    sum = sum + sgn*npwr/nfac
    sgn = -sgn
    # print("logp1cer sum=",sum)

  # end i
  # print("logp1cer sum=",sum)
  # print(" ")
  return sum

# end logp1ser

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

  # end i
  return float(fct)

# end factorial

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

  # end i
  return pwr

# end power

print("trig_series.py3 running")
x = 2.0
n = 10
expv = expser(x,n)
print("expser(",x,",",n,")=",expv)
expa = math.exp(x)
print("math.exp(",x,")=",expa)
print("expser error=",abs(expv-expa))
print(" ")
sinv = sinser(x,n)
print("sinser(",x,",",n,")=",sinv)
sina = math.sin(x)
print("math.sin(",x,")=",sina)
print("sinser error=",abs(sinv-sina))
print(" ")
cosv = cosser(x,n)
print("cosser(",x,",",n,")=",cosv)
cosa = math.cos(x)
print("math.cos(",x,")=",cosa)
print("cosser error=",abs(cosv-cosa))
print(" ")
sinhv = sinhser(x,n)
print("sinhser(",x,",",n,")=",sinhv)
sinha = math.sinh(x)
print("math.sinh(",x,")=",sinha)
print("sinhser error=",abs(sinhv-sinha))
print(" ")
coshv = coshser(x,n)
print("coshser(",x,",",n,")=",coshv)
cosha = math.cosh(x)
print("math.cosh(",x,")=",cosha)
print("coshser error=",abs(coshv-cosha))
print(" ")
logv = logser(x,n)
print("logser(",x,",",n,")=",logv)
loga = math.log(x)
print("math.log(",x,")=",loga)
print("logser error=",abs(logv-loga))
print(" ")
x = 1.0
logp1v = logp1ser(x,n)
print("logp1ser(",x,",",n,")=",logp1v)
logp1a = math.log(x+1.0)
print("math.log(",x,"+1)=",logp1a)
print("logp1ser error=",abs(logp1v-logp1a))
print(" ")
print("trig_series.py3 finished")