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Sec

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Sec[z]

gives the secant of z (= 1 / Cos[z]).

Notes Sec is Listable. Singularities at z = Pi/2 + k Pi yield ComplexInfinity.

Examples (8)

Every input below was run against the current Mathilda build and its output recorded.

Applications (8)

In[1]:= Sec[Pi/3]
Out[1]= 2

In[2]:= Sec[0]
Out[2]= 1

In[3]:= N[Sec[1]]
Out[3]= 1.85082

In[4]:= Sec[Pi/4]
Out[4]= Sqrt[2]

In[5]:= Sec[Pi/5]
Out[5]= -1 + Sqrt[5]

In[6]:= Sec[I]
Out[6]= Sech[1]

In[7]:= Series[Sec[x], {x, 0, 6}]
Out[7]= 1 + 1/2 x^2 + 5/24 x^4 + 61/720 x^6 + O[x]^7

In[8]:= N[Sec[1], 40]
Out[8]= 1.8508157176809256179117532413986501934704

Implementation notes

Algorithm. builtin_sec follows the src/trig.c cascade but uses even symmetry: strip_inverse_call(arg, "ArcSec") for Sec[ArcSec[x]] -> x; even_fold for Sec[-x] -> Sec[x] when the argument is superficially negative; trig_i_fold(arg, "Sech", 0) for Sec[I y] -> Sech[y]; and Sec[0] = 1. Exact values at rational multiples of Pi are recognised by extract_pi_multiplier and produced by exact_sec.

Numeric. MPFR arguments use numeric_mpfr_apply_unary(..., mpfr_sec) (complex fallback mpfr_complex_sec); otherwise get_approx computes 1.0 / ccos(c) for inexact inputs, yielding EXPR_REAL or Complex. Symbolic input returns NULL. Attributes: ATTR_LISTABLE | ATTR_NUMERICFUNCTION | ATTR_PROTECTED.

Attributes: Listable, NumericFunction, Protected.

References

Notes & additional examples

Notes

Sec[z] is 1/Cos[z]. Singularities at z = Pi/2 + k Pi yield ComplexInfinity. Sec is Listable.