模組:Complex Number/Functions

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本模組定義了一些可供Module:Complex_Number系列模組使用的擴充函數。

使用條件

要能使用此函數，必須先輸入一個數字類別資料結構以及其專用的math程式庫

數字類別資料結構必須包含下列成員函數：
- clean() : 削去過小的小數，如5.0000000000000015會被削成5。
- metatable必須實作下列函數
  加法、減法、乘法、除法、取餘數、相等判斷、轉字串、相反數（加法反元素）
專用的math程式庫必須包含下列成員：
- 基本函數：
  abs（絕對值）、 sgn（符號函數）、 arg（輻角）、 conjugate（共軛）、 sqrt（平方根）、 dot（內積）
- 高斯符號：
  floor（無條件捨去）、 ceil（無條件進位）、 round（四捨五入）
- 除法相關：
  mod、 div（除法）、 inverse（乘法反元素）
- 指對數相關：
  exp（自然指数）、 log（自然對數）、 pow（冪）
- 特殊成員
  re（返回實部）、nonRealPart（返回不含實部的複數）、 tovector（返回複數的數組形式）
- 三角函數
  sin、 cos、 tan、 cot、 cis、 asin、 acos、 atan、 acot
- 雙曲函數
  sinh、 cosh、 tanh、 coth、 asinh、 acosh、 atanh、 acoth
- 常數
  e、 pi（圓周率）、 nan（不是數字）

使用方法

初始化任何符合此擴充函數庫使用條件的數學庫
- local 自訂函數庫名稱 = require("Module:Complex Number").函數庫名稱.init()
  以Module:Complex Number的cmath為例：
  local cmath = require("Module:Complex Number").cmath.init()
初始化本擴充函數庫
- 自訂函數庫名稱 = require("Module:Complex Number/Functions")._init(自訂函數庫名稱, 函數庫對應的數字建構函式)
  以上述之Module:Complex Number的cmath為例：
  cmath = require("Module:Complex Number/Functions")._init(cmath, cmath.constructor)
使用擴充函數庫中的函數
例如：
print(cmath.factorial(5), cmath.sec(cmath.pi/4))

輸出：120 　　1.4142135623731

模組中的函數

三角函數擴充

擴充了原本未定義的三角函數

如sec（正割）、 csc（餘割）、 sech（雙曲正割）、 csch（雙曲餘割）、 asec（反正割）、 acsc（反餘割）、asech（反雙曲正割）、 acsch（反雙曲餘割）、 gd（古德曼函數）、 cogd（餘古德曼函數）、 arcgd（反古德曼函數）

功能: 輸入一個複數x，回傳其指定三角函數的值

range(x,min,max)

功能: 只取函數的某一段; 若x位於min,max區間內，則回傳x，否則回傳NaN

統計函數

定義了一些統計函數

如minimum（最小值）、 maximum（最大值）、 average（平均值）、 geoaverage（幾何平均值）、 var（變異數）、 σ（標準差）

功能: 輸入一系列數字，回傳其指定的統計值

diff(function, x₀)

功能: 輸入一個函數，計算該函數在x=x₀的導數
實作方式: 數值微分#高階方法

integral(a, b, function, step)

功能: 輸入一個函數，計算從a到b的定積分，並以step為求黎曼和的間距
實作方式: en:Boole's_rule

limit(x₀, way, function)

功能: 輸入一個函數，計算從way方向向x₀逼近的極限。; 其中，way=1為右極限、way=-1為左極限、way=0為不分方向的極限，若左極不等於右極回傳NaN

條件式

常數條件輸入

if(條件, 為真時, 為假時)、ifelse(條件1, 條件1為真, 條件2, 條件2為真, ... ,皆為假)

代表條件在傳入函數時已經完成計算

函數條件輸入

iff(條件函數, 為真時, 為假時)、ifelsef(條件函數1, 條件1為真, 條件函數2, 條件2為真, ... ,皆為假)

代表條件在傳入函數時尚未計算，判斷的當下才計算。所傳入的函數需要是無參數函數，若有參數也只會被忽略。用於定義遞迴下的條件

factorial(x)

功能: 輸入一個複數x，回傳其階乘值; 即factorial(x) $=n!$
實作方式: 參考#gamma(x)

binomial(n,k)

功能: 計算二項式係數; 也可以理解為從n個元素中取出k個元素的方法數

gcd(a,b,c,...)

功能: 計算a,b,c,....等數字的最大公因數，支援複數。
實作方式: 輾轉相除法

lcm(a,b,c,...)

功能: 計算a,b,c,....等數字的最小公倍數，支援複數。
實作方式: 最小公倍數#计算方法

gamma(x)

功能: 輸入一個複數x，回傳其Γ函數值

精確度: 有效數字14位
運算效率: 平均一次運算耗時約0.3582毫秒（3.6×10⁻⁴ s、一秒可計算2,700+次），測試於2018年11月19日 (一) 06:39 (UTC)、2022年4月12日 (二) 17:54 (UTC)。
實作方式

共分成4個部分
- 中間藍色部分是利用從零展開倒數伽瑪函數的泰勒級數定義
  展開至前30項
  
  ${\frac {1}{\Gamma (z)}}=z+\gamma z^{2}+\left({\frac {\gamma ^{2}}{2}}-{\frac {\pi ^{2}}{12}}\right)z^{3}+\cdots$
  
  $a_{n}={\frac {a_{2}a_{n-1}-\sum _{j=2}^{n-1}(-1)^{j}\,\zeta (j)\,a_{n-j}}{n-1}}$ ^[1]
- 兩側橘紅色部分是利用中間藍色代Γ函數的遞迴關係式定義，並用For迴圈實作
  $\Gamma (x+1)=x\Gamma (x)$
  
  $\Gamma (x-1)={\frac {\Gamma (x)}{x-1}}$
- 上下的綠色部分則是使用Robert H. Windschitl (2002) 所提出的公式近似
  $\Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {z}{e}}{\sqrt {z\sinh {\frac {1}{z}}+{\frac {1}{810z^{6}}}}}\right)^{z}$ ^[2]
- 最後黃色部分則是使用帶有斯特靈級數的斯特靈公式近似
  $n!={\sqrt {2\pi n}}\left({n \over e}\right)^{n}\left(1+{1 \over 12n}+{1 \over 288n^{2}}-{139 \over 51840n^{3}}-{571 \over 2488320n^{4}}+\cdots \right).$ ^[3]
  
  展開至前16項（來源：（OEIS數列A001163）、（OEIS數列A001164））
- 而背景透明標記（灰白相間）部分則為超出浮點數可儲存範圍，會溢位或出現inf或nan
- 最左邊土黃色則是可能出現低於設計的精確度小數12位而回傳0

參考文獻

^ Wrench, J.W. (1968). Concerning two series for the gamma function. Mathematics of Computation, 22, 617–626. and
Wrench, J.W. (1973). Erratum: Concerning two series for the gamma function. Mathematics of Computation, 27, 681–682.
^ Viktor T. Toth. "Programmable Calculators: Calculators and the Gamma Function". 2006. （原始内容存档于2007-02-23）.
^ F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds. NIST Digital Library of Mathematical Functions.

上述文档嵌入自Module:Complex Number/Functions/doc。 (编辑 | 历史)
编者可以在本模块的沙盒 (编辑 | 差异)和测试样例 (创建)页面进行实验。
本模块的子页面。

local p = {}
local math_lib
local to_number
function p._init(_math_lib, _to_number)
	local warp_funcs={"factorial","gamma","sec","csc","sech","csch","asec","acsc","asech","acsch","gd","cogd","arcgd",
		"LambertW","norm",
		"gcd","lcm","range","binomial",'minimum','maximum','average','min','max','avg','geoaverage','var','σ',
		'selectlist','for','while','summation','product','if','iff','ifelse','ifelsef','diff','integral', '∫', 'limit',
		'hide','exprs','lastexpr','equalexpr',--調整條目中定義不顯示的函數
		'randomseed','time','nil','null','call',
		'frameArg','object','string','symbols','passObject','typeof','length','array','assignMember',
		'divisorsigma','findnext','findlast','divisor','primedivisor','eulerphi'
	}
	for i=1,#warp_funcs do
		if _math_lib[ warp_funcs[i] ] == nil then
			_math_lib[ warp_funcs[i] ] = p['_' .. warp_funcs[i] ]
		end
	end
	math_lib = _math_lib
	to_number = _to_number
	return _math_lib
end
function p._complex_number()
	return p._init(require("Module:Complex Number").cmath.init(), require("Module:Complex Number").cmath.init().toComplexNumber)
end

local noop_func = function()end
local function assertArg(val, index, func)
	assert (val ~= nil, string.format(
		"bad argument #%d to '%s' (number expected, got %s)", index, func, type(val)))
end
function p._binomial(cal_n,cal_k)
	local r_n=tonumber(tostring(cal_n))
	local r_k=tonumber(tostring(cal_k))
	if r_n and r_k then
		if r_n>0 and r_k>=0  then
			local f_n, f_k;
			_,f_n = math.modf(r_n);_,f_k = math.modf(r_k)
			if math.abs(f_n) < 1e-12 and math.abs(f_k) < 1e-12 then
				local result = 1
				if r_n == 0 then return result end
				while r_k>0 do
					result = result * r_n / r_k
					r_n = r_n - 1
					r_k = r_k - 1
				end
				return result
			end
		end
	end
	local n=to_number(cal_n)
	local k=to_number(cal_k)
	assertArg(n, 1, 'binomial') 
	assertArg(k, 2, 'binomial')
	return p._factorial(n) * math_lib.inverse( p._factorial(k) ) * math_lib.inverse( p._factorial(n-k) )
end

function p._factorial(cal_z)return p._gamma(to_number(cal_z) + 1)end
function p._sec(cal_z)return math_lib.inverse( math_lib.cos( to_number(cal_z) ) )end
function p._csc(cal_z)return math_lib.inverse( math_lib.sin( to_number(cal_z) ) )end
function p._sech(cal_z)return math_lib.inverse( math_lib.cosh( to_number(cal_z) ) )end
function p._csch(cal_z)return math_lib.inverse( math_lib.sinh( to_number(cal_z) ) )end
function p._asec(cal_z)return  math_lib.acos( math_lib.inverse( to_number(cal_z) ) )end
function p._acsc(cal_z)return  math_lib.asin( math_lib.inverse( to_number(cal_z) ) )end
function p._asech(cal_z)return  math_lib.acosh( math_lib.inverse( to_number(cal_z) ) )end
function p._acsch(cal_z)return  math_lib.asinh( math_lib.inverse( to_number(cal_z) ) )end
function p._gd(cal_z)return math_lib.atan( math_lib.tanh( to_number(cal_z) * 0.5 ) ) * 2 end
function p._arcgd(cal_z)return math_lib.atanh( math_lib.tan( to_number(cal_z) * 0.5 ) ) * 2 end
function p._cogd(cal_z)local x = to_number(cal_z); return - math_lib.sgn( x ) * math_lib.log( math_lib.abs( math_lib.tanh( x * 0.5 ) ) ) end

function p._range(val,vmin,vmax)
	assertArg(val, 1, 'range')
	local min_inf, max_inf = tonumber("-inf"), tonumber("inf")
	local function get_vector(_val)
		local val = to_number(_val)
		if val == nil then return {} end
		return math_lib.tovector(val)
	end
	local v_val, v_min, v_max = math_lib.tovector(to_number(val)), get_vector(vmin), get_vector(vmax)
	for i=1,#v_val do
		local min_v, max_v = math.min(v_min[i]or min_inf, v_max[i]or max_inf), math.max(v_min[i]or min_inf, v_max[i]or max_inf)
		if v_val[i] < min_v or v_val[i] > max_v then return to_number(math_lib.nan) end
	end
	return to_number(val)
end
function p._calc_diff(_value, _left, _right, _expr)
	local value = to_number(_value)
	local left = to_number(_left)
	local right = to_number(_right)
	local left_val = _expr(value + left)
	local right_val = _expr(value + right)
	return (left_val - right_val) / (left - right)
end
function p._diff_abramowitz_stegun(_x, _h, _f)
	local x = to_number(_x)
	local h = to_number(_h)
    local _1 = to_number(1) local _2 = to_number(2) local _8 = to_number(8) local _12 = to_number(12)
    --if tonumber(math_lib.abs(math_lib.nonRealPart(x))) > 1e-8 then
    --	--[[數值微分#複變的方法]] --(Martins, Sturdza et.al.)
    --	local _i = math_lib.i or _1
    --	return (math_lib.im(_f(x + _i * h))) / h
    --else
    	--[[數值微分#高階方法]] --(Abramowitz & Stegun, Table 25.2)
		return (- _f(x + _2*h) + _8*_f(x + h) - _8*_f(x - h) + _f(x - _2*h)) / (_12 * h)
	--end
end

function p._calc_from(_value, _left, _right, _expr)
	local value = to_number(_value)
	local left = to_number(_left)
	local left_val = _expr(value + left)
	return left_val - p._calc_diff(_value, _left, _right, _expr) * left
end
function p._get_sample_number(_value)
	local num = math_lib.re(math_lib.abs(to_number(_value)))
	return (tonumber(num)~=nil) and ((num < 1e-8) and 1 or num) or 1
end
function p._diff(_expr, _value)
	local func_type = type(noop_func)
	local value_str = tostring(_value)
	if value_str == 'nan' or value_str == '-nan' or value_str == 'nil'then
		error("計算失敗：無效的數值 ")
		return _expr
	end
	if value_str == 'inf' or value_str == '-inf'then
		error("計算失敗：不支援無窮微分 ")
		return _expr
	end
	local small_scale_pow = math_lib.floor(math_lib.log(p._get_sample_number(_value)) / math_lib.log(to_number(10)))-5
	local small_scale = math_lib.pow(10,small_scale_pow)
	if type(_expr) == func_type then
		return p._diff_abramowitz_stegun(_value, small_scale, _expr)
	else
		return 0
	end
end

function p._integral(_a,_b,func,step)
	local a = to_number(_a)
	local b = to_number(_b)
	local begin_str = tostring(_a)
	local stop_str = tostring(_b)
	if begin_str == 'nan' or begin_str == '-nan' or begin_str == 'nil' or stop_str == 'nan' or stop_str == '-nan' or stop_str == 'nil'then
		error("計算失敗：無效的迴圈 ")
		return func
	end
	if begin_str == 'inf' or begin_str == '-inf' or stop_str == 'inf' or stop_str == '-inf'then
		error("計算失敗：不支援無窮求積 ")
		return func
	end
	local n = tonumber(step) or 2000
	local f = (type(func)==type(noop_func))and func or (function()return to_number(func) or 0 end)
	local h = (b-a)/n
	local x0, xn = a, b
	local _2, _7, _12, _14, _32, _45 = 
		to_number(2), to_number(7), to_number(12), to_number(14), to_number(32), to_number(45)
	local i0, i1, i2 = 1, 2, 4
	local sumfxi0, sumfxi1, sumfxi2 = to_number(0), to_number(0), to_number(0)
	for i=1,n do -- Boole's rule
		if i0 > n-1 and i1 > n-2 and i2 > n-4 then break end
		local xi0, xi1, xi2 = a + i0 * h, a + i1 * h, a + i2 * h
		if i0 <= n-1 then sumfxi0 = sumfxi0 + f(xi0) end
		if i1 <= n-2 then sumfxi1 = sumfxi1 + f(xi1) end
		if i1 <= n-4 then sumfxi2 = sumfxi2 + f(xi2) end
		i0 = i0 + 2
		i1 = i1 + 4
		i2 = i2 + 4
	end
	return (_2 * h / _45) * (_7 * (f(x0) + f(xn)) + _32 * sumfxi0 + _12 * sumfxi1 + _14 * sumfxi2)
end
p['_∫'] = p._integral

function p._limit(_value, _way, _expr)
	local way = to_number(_way)
	local func_type = type(noop_func)
	local value_str = tostring(_value)
	if value_str == 'nan' or value_str == '-nan' or value_str == 'nil' then
		error("計算失敗：無效的數值 ")
		return _expr
	end
	if value_str == 'inf' or value_str == '-inf'then
		error("計算失敗：不支援無窮極限 ")
		return _expr
	end
	local small_scale_pow = tonumber(math_lib.re(math_lib.floor(math_lib.log(p._get_sample_number(_value)) / math_lib.log(to_number(10))))-6)
	local small_scale = math.pow(10,small_scale_pow)
	local small_scale_a = small_scale / 10
	local small_scale_c = small_scale * 10

	if type(_expr) == func_type then
		if math_lib.re(math_lib.abs(way)) < 1e-10 then
			local left = p._calc_from(_value, -small_scale, -small_scale_a, _expr)
			local right = p._calc_from(_value, small_scale_a, small_scale, _expr)
			if math_lib.re(math_lib.abs(left - right)) < small_scale_c then
				return (left + right) / 2
			else
				return math_lib.nan
			end
		else
			return (math_lib.re(way) > 0) and p._calc_from(_value, small_scale_a, small_scale, _expr) or
				p._calc_from(_value, -small_scale, -small_scale_a, _expr)
		end
	else
		return _expr
	end
end
p._nil = "nil"
p._null = "nil"
---------------------- 流程控制擴充 ----------------------
function p._if(_expr, _true_expr, _false_expr)
	return p._ifelse_func(false, _expr, _true_expr, _false_expr)
end
function p._iff(_expr, _true_expr, _false_expr)
	return p._ifelse_func(true, _expr, _true_expr, _false_expr)
end
function p._ifelse(...)
	return p._ifelse_func(false, ...)
end
function p._ifelsef(...)
	return p._ifelse_func(true, ...)
end
function p._ifelse_func(is_func, ...)
	local func = noop_func
	local exprlist = {...}
	local last_else = #exprlist % 2 == 1
	local max_num = (last_else and (#exprlist - 1) or #exprlist) / 2
	for i=1,max_num do
		local _expr = exprlist[i * 2 - 1]
		local expr = (type(_expr) == type(func)) and _expr() or _expr
		local expr_true = exprlist[i * 2]
		local _chk_flag = math_lib.abs(to_number(expr)) > 1e-14;
		if _chk_flag then 
			return (type(expr_true) == type(func) and is_func) and expr_true() or expr_true 
		end
	end
	if last_else then
		local expr_false = exprlist[#exprlist]
		return (type(expr_false) == type(func) and is_func) and expr_false() or expr_false 
	end
	local _expr = exprlist[1]
	return (type(_expr) == type(func)) and _expr() or _expr
end
local function check_while(_ifexpr)
	local result = (type(_ifexpr) == type(noop_func)) and _ifexpr() or _ifexpr
	if result == true then return true end
	if not result then return false end
	return math_lib.abs(to_number(result)) > 1e-14
end
function p._while(_ifexpr, _expr)
	local result
	while check_while(_ifexpr) do
		result = (type(_expr) == type(noop_func)) and _expr() or _expr
		if type(result) == type({}) and result['return'] then break end
	end
	return result
end
function p._for(_start,_end,_step,_expr)
	local _begin = to_number(_start);
	local _stop = to_number(_end);
	local _do_step = to_number(_step);
	local check_loop = (_stop - _begin) / _do_step
	local begin_str = tostring(_begin)
	local stop_str = tostring(_stop)
	if math_lib.re(math_lib.abs(_do_step))<=1e-14 or math_lib.re(check_loop) < 0 or
		begin_str == 'nan' or begin_str == '-nan' or begin_str == 'nil' or begin_str == 'inf' or begin_str == '-inf' or 
		stop_str == 'nan' or stop_str == '-nan' or stop_str == 'nil' or stop_str == 'inf' or stop_str == '-inf'then 
			error("計算失敗：無效的迴圈 ")
			return _expr
	end
	if type(_expr) == type(noop_func) then
		local it = _begin
		local init = to_number(0)
		while math_lib.re(it) <= math_lib.re(_stop) do
			init = _expr(to_number(it),to_number(init))
			if type(init) == type({}) and init['return'] then break end
			it = it + _do_step
		end
		return init
	else
		return _expr
	end
end
function p._summation(_start,_end,_expr)
	local _begin = to_number(_start);
	local _stop = to_number(_end);
	local _do_step = to_number(1);
	local check_loop = (_stop - _begin) / _do_step
	local begin_str = tostring(_begin)
	local stop_str = tostring(_stop)
	if math_lib.re(math_lib.abs(_do_step))<=1e-14 or math_lib.re(check_loop) < 0 or 
		begin_str == 'nan' or begin_str == '-nan' or begin_str == 'nil' or stop_str == 'nan' or stop_str == '-nan' or stop_str == 'nil'then
			error("計算失敗：無效的迴圈 ")
			return _expr
		end
	if begin_str == 'inf' or begin_str == '-inf' or stop_str == 'inf' or stop_str == '-inf'then
		error("計算失敗：不支援無窮求和 ")
		return _expr
	end
	local func_type = type(noop_func)
	local it = _begin
	local init = to_number(0)--空和
	while math_lib.re(it) <= math_lib.re(_stop) do
		init = init + ((type(_expr) == func_type) and _expr(to_number(it)) or to_number(_expr))--累加
		it = it + _do_step
	end
	return init
end

function p._product(_start,_end,_expr)
	local _begin = to_number(_start);
	local _stop = to_number(_end);
	local _do_step = to_number(1);
	local check_loop = (_stop - _begin) / _do_step
	local begin_str = tostring(_begin)
	local stop_str = tostring(_stop)
	if math_lib.re(math_lib.abs(_do_step))<=1e-14 or math_lib.re(check_loop) < 0 or 
		begin_str == 'nan' or begin_str == '-nan' or begin_str == 'nil' or stop_str == 'nan' or stop_str == '-nan' or stop_str == 'nil'then
			error("計算失敗：無效的迴圈 ")
			return _expr
	end
	if begin_str == 'inf' or begin_str == '-inf' or stop_str == 'inf' or stop_str == '-inf'then
		error("計算失敗：不支援無窮求積 ")
		return _expr
	end
	local func_type = type(noop_func)
	local it = _begin
	local init = to_number(1)--空積
	while math_lib.re(it) <= math_lib.re(_stop) do
		init = init * ((type(_expr) == func_type) and _expr(to_number(it)) or to_number(_expr))--累乘
		it = it + _do_step
	end
	return init
end
---------------------- 工具函數擴充 ----------------------
function p._randomseed(_seed)
	local seed = tonumber(tostring(_seed)) or (os.time() * os.clock())
	math.randomseed(math.floor(seed))
	return to_number(seed)
end
function p._time() return to_number(os.time()) end
function p._call(func, ...)
	if type(func) == type(noop_func) then return func(...)end
	return func
end

function p._hide(...)
	local input_args = {...}
	return to_number(input_args[#input_args])
end
p._exprs = p._hide
p._lastexpr = p._hide
p._equalexpr = p._hide

---------------------- 統計函數 ----------------------
function p._selectlist(x,...) 
	local input_args = {...}
	local y = input_args[1]
	local z = input_args[2]
	local id_x = tonumber(tostring(x)) or 0
	if type(y) == type("string") then return mw.ustring.sub(y,id_x,id_x) end
	if type(y)==type({}) and #y >= id_x and id_x>0 then
		if id_x <= 0 then id_x = id_x + #y + 1 end
		return y[id_x] or tonumber('nan') 
	elseif type(z)==type({}) and #z >= id_x then
		local id_y = tonumber(tostring(y)) or 0
		if id_x <= 0 then id_x = id_x + #z + 1 end
		if type(z[id_x])==type({}) and #(z[id_x]) >= id_y and id_y>0 then
			if id_y <= 0 then id_y = id_y + #(z[id_x]) + 1 end
			return (z[id_x][id_y]) or tonumber('nan')
		end
	end
	id_x = tonumber(tostring(x)) or 0
	if id_x <= 0 then id_x = id_x + #input_args + 1 end
	return input_args[id_x] or tonumber('nan') 
end
function p._minimum(...) return p.minmax('min', ...) end
function p._maximum(...) return p.minmax('max', ...) end
function p._average(...) return p.minmax('avg', ...) end
function p._geoaverage(...) return p.minmax('gavg', ...) end
function p._var(...) return p.minmax('var', ...) end
p._min = p._minimum
p._max = p._maximum
p._avg = p._average
p['_σ'] = function(...) return p.minmax('σ',...) end
local function flatten(inarray,outarray)
	outarray = outarray or {}
	if type(inarray) ~= type({}) then
		outarray[#outarray + 1] = inarray
	elseif inarray.numberType then
		outarray[#outarray + 1] = inarray
	elseif type(inarray.args) == type({}) then
		local midarray = inarray.args
		for k,v in pairs(midarray) do
			local i = tonumber(k)
			if i then outarray = flatten(midarray[i],outarray)end
		end
		if type(inarray.getParent) == type(noop_func) then
			midarray = (inarray:getParent() or {}).args or {}
			for k,v in pairs(midarray) do
				local i = tonumber(k)
				if i then outarray = flatten(midarray[i],outarray)end
			end
		end
	elseif #inarray > 0 then
		for i=1,#inarray do outarray = flatten(inarray[i],outarray)end
	end
	return outarray
end
function p.minmax(calc_mode,...)
	local mode = calc_mode
	local tonumber_lib = to_number or tonumber
	local lib_math = math_lib or math
	local args, tester = flatten({ ... }), {tonumber("nan")}
	if type(calc_mode) == type({}) then mode = (calc_mode.args or calc_mode).mode or mode; args = flatten({args, calc_mode}) end
	local sum, prod, count, sumsq, sig = tonumber_lib(0), tonumber_lib(1), 0, tonumber_lib(0), (mode =='var'or mode=='σ')
	local mode_map = {}
	local non_nan
	for i=1,#args do
		local got_number, calc_number = tonumber(tostring(args[i])) or tonumber("nan"), tonumber_lib(args[i])
		if calc_number then sum, prod, count = calc_number + sum, calc_number * prod, count + 1 end
		if sig == true then 
			local x_2 = calc_number * calc_number
			if lib_math.dot then
				x_2 = lib_math.dot(calc_number, lib_math.conjugate(calc_number))
			end
			sumsq = sumsq + x_2 
		end
		mode_map[args[i]] = (mode_map[args[i]]or 0) + 1
		if tostring(got_number):lower()~="nan" and type(non_nan) == type(nil) then
			tester[1], non_nan = got_number, got_number
		else tester[#tester + 1] = got_number end
	end
	local modes={min=math.min,max=math.max,sum=function()return sum end,prod=function()return prod end,count=function()return count end,
		avg=function()return sum*tonumber_lib(1/count) end,
		gavg=function()return lib_math.pow(prod,tonumber_lib(1/count)) end,
		var=function()return sumsq*tonumber_lib(1/count)-sum*sum*tonumber_lib(1/(count*count)) end,
		['σ']=function()return lib_math.sqrt(sumsq*tonumber_lib(1/count)-sum*sum*tonumber_lib(1/(count*count))) end,
		mode=function()
			local max_count, mode_data = 0, ''
			for mkey, mval in pairs(mode_map) do
				if mval > max_count then
					max_count = mval
					mode_data = mkey
				end
			end
			return mode_data
		end,
		gcd=function(...)if not to_number or not math_lib then p._complex_number()end return p._gcd(...)end,
		lcm=function(...)if not to_number or not math_lib then p._complex_number()end return p._lcm(...)end,
	}
	if tostring(tester[1]):lower()=="nan" and mode:sub(1,1)=='m' then
		local error_msg = ''
		for i=1,#args do if error_msg~=''then error_msg = error_msg .. '、 ' end
			error_msg = error_msg .. tostring(args[i])
		end
		error("給定的數字 " .. error_msg .." 無法比較大小") 
	end
	if type(modes[mode]) ~= type(tonumber) then
		error("未知的統計方式 '" .. mode .."' ") 
	end
	return modes[mode](unpack(tester))
end

local function fold(func, ...)
	-- Use a function on all supplied arguments, and return the result. The function must accept two numbers as parameters,
	-- and must return a number as an output. This number is then supplied as input to the next function call.
	local vals = {...}
	local count = #vals -- The number of valid arguments
	if count == 0 then return
		-- Exit if we have no valid args, otherwise removing the first arg would cause an error.
		nil, 0
	end 
	local ret = table.remove(vals, 1)
	for _, val in ipairs(vals) do
		ret = func(ret, val)
	end
	return ret, count
end

--[[
Fold arguments by selectively choosing values (func should return when to choose the current "dominant" value).
]]
local function binary_fold(func, ...)
	local value = fold((function(a, b) if func(a, b) then return a else return b end end), ...)
	return value
end
---------------------- 伽瑪函數 ----------------------
local Reciprocal_gamma_coeff = {1,0.577215664901532860607,-0.655878071520253881077,-0.0420026350340952355290,0.166538611382291489502,-0.0421977345555443367482,-0.00962197152787697356211,0.00721894324666309954240,-0.00116516759185906511211,-0.000215241674114950972816,0.000128050282388116186153,-0.0000201348547807882386557,-1.25049348214267065735e-6,1.13302723198169588237e-6,-2.05633841697760710345e-7,6.11609510448141581786e-9,5.00200764446922293006e-9,-1.18127457048702014459e-9,1.04342671169110051049e-10,7.78226343990507125405e-12,-3.69680561864220570819e-12,5.10037028745447597902e-13,-2.05832605356650678322e-14,-5.34812253942301798237e-15,1.22677862823826079016e-15,-1.18125930169745876951e-16,1.18669225475160033258e-18,1.41238065531803178156e-18,-2.29874568443537020659e-19,1.71440632192733743338e-20}
--https://oeis.org/A001163 、 https://oeis.org/A001164
local stirling_series_coeff = {1,0.0833333333333333333333333,0.00347222222222222222222222,-0.00268132716049382716049383,-0.000229472093621399176954733,0.000784039221720066627474035,0.0000697281375836585777429399,-0.000592166437353693882864836,-0.0000517179090826059219337058,0.000839498720672087279993358,0.0000720489541602001055908572,-0.00191443849856547752650090,-0.000162516262783915816898635,0.00640336283380806979482364,0.000540164767892604515180468,-0.0295278809456991205054407,-0.00248174360026499773091566,0.179540117061234856107699,0.0150561130400264244123842,-1.39180109326533748139915,-0.116546276599463200850734}
function p._gamma_high_imag(cal_z)
	local z = to_number(cal_z)
	if z ~= nil and math_lib.abs(math_lib.nonRealPart(z)) > 2 then
		local inv_z = math_lib.inverse(z)
		return math_lib.sqrt((math_lib.pi * 2) * inv_z) * math_lib.pow(z * math_lib.exp(-1) *
			math_lib.sqrt( (z * math_lib.sinh(inv_z) ) + math_lib.inverse(to_number(810) * z * z * z * z * z * z) ),z)
	end
	return nil
end
function p._gamma_morethen_lua_int(cal_z)
	local z = to_number(cal_z) - to_number(1)
	local lua_int_term = 18.1169 --FindRoot[ Factorial[ x ] == 2 ^ 53, {x, 20} ]
	if math_lib.abs(z) > (lua_int_term - 1) or (math_lib.re(z) < 0 and math_lib.abs(math_lib.nonRealPart(z)) > 1 ) then
		local sum = 1
		for i = 1, #stirling_series_coeff - 1 do
			local a, n = to_number(z), tonumber(i) local y, k, f = to_number(1), n, to_number(a)
			while k ~= 0 do 
				if k % 2 == 1 then y = y * f end 
				k = math.floor(k / 2); f = f * f
			end
			sum = sum + stirling_series_coeff[i + 1] * math_lib.inverse(y)
		end
		return math_lib.sqrt( (2 * math.pi) * z ) * math_lib.pow( z * math.exp(-1), z ) * sum
	end
	return nil
end
function p._gamma_abs_less1(cal_z)
	local z = to_number(cal_z)
	if (math.abs(math_lib.re(z)) <=1.001) then
		if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ( (math.abs(math_lib.re(z) - 1) < 1e-14) or (math.abs(math_lib.re(z) - 2) < 1e-14) ) then return to_number(1)end
		return math_lib.inverse(p._recigamma_abs_less1(z))
	end
	return nil
end
function p._recigamma_abs_less1(z)
	local result = to_number(0)
	for i=1,#Reciprocal_gamma_coeff do
		result = result + Reciprocal_gamma_coeff[i] * math_lib.pow(z,i)
	end
	return result
end
function p._gamma(cal_z)
	local z = to_number(cal_z)
	if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ((math_lib.re(z) < 0 or math.abs(math_lib.re(z)) < 1e-14)
		and math.abs(math.floor(math_lib.re(z)) - math_lib.re(z)) < 1e-14 ) then return tonumber("nan") end
	local pre_result = p._gamma_morethen_lua_int(z) or p._gamma_high_imag(z) or p._gamma_abs_less1(z)
	if pre_result then return pre_result end
	local real_check = math_lib.re(z)
	local loop_count = math.floor(real_check)
	local start_number, zero_flag = z - loop_count, false
	if math_lib.abs(start_number) <= 1e-14 then start_number = to_number(1);zero_flag = true end
	local result = math_lib.inverse(p._recigamma_abs_less1(start_number))
	if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ((math_lib.re(z) > 1e-14 )and math.abs(math.floor(math_lib.re(z)) - math_lib.re(z)) < 1e-14 ) then result = to_number(1)  end
	local j = to_number(start_number)
	for i=1,math.abs(loop_count) do
		if loop_count > 0 then result = result * j else result = result * math_lib.inverse(j-1) end
		if zero_flag==true and loop_count > 0 then zero_flag=false else if loop_count > 0 then j = j + 1 else j = j - 1 end end
	end
	if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ((math_lib.re(z) > 1e-14 )and math.abs(math.floor(math_lib.re(z)) - math_lib.re(z)) < 1e-14 ) then return math_lib.floor(result) end
	return result
end

---------------------- 最大公因數與最小公倍數 ----------------------
local function findGcd(a, b)
	local r, oldr = to_number(b), to_number(a)
	while math_lib.abs(r) > 1e-6 do local mod_val = oldr % r oldr, r = to_number(r), mod_val end
	if math_lib.abs(math_lib.nonRealPart(oldr)) < 1e-14 and (math_lib.re(oldr) < 0 ) then oldr = -oldr end
	return oldr
end

function p._gcd(...)
	local result, count = fold(findGcd, ...)
	return result
end

function p._lcm(...)
	local function findLcm(_a, _b)
		local a, b = to_number(_b), to_number(_a)
		return math_lib.abs(a * b) / findGcd(a, b)
	end
	local result, count = fold(findLcm, ...)
	return result
end

---------------------- 字串與物件擴充 (提供Module:Complex Number/Calculate使用) ----------------------
function p._symbols(name)
	return({
		comma = ',', space = ' ', colon = ':', dot = '.', squot = "'", dquot = '"', semicolon = ';', underline = '_',
		lcbracket = '{', rcbracket = '}', lsbracket = '[', rsbracket = ']', lpbracket = '(', rpbracket = ')',
		plus = '+', minus = '-', mul = '*', div = '/', ['pow'] = '^', equal = '=',
		lt = '<', gt = '>', money = '$', percent = '%', ['and'] = '&', exclamation = '!', at = '@', hashtag = '#', to = '~', slash ='\\'
	})[name]
end

function p._frameArg(str)
	local frame = mw.getCurrentFrame()
	local working_frame = frame:getParent() or frame
	local argname = tonumber(tostring(str)) or tostring(str)
	return working_frame.args[argname] or frame.args[argname]
end

function p._string(str,...)
	local result = tostring(str)
	local str_list = {...}
	for i = 1,#str_list do
		result = result .. '' .. tostring(str_list[i])
	end
	return result
end

function p._passObject(obj)
	return obj
end

function p._assignMember(obj, member, value)
	local input_obj = obj
	if type(obj) == type("string") then input_obj = _G[obj] end
	if type(obj) == type(0) or type(input_obj) == type(0) then error("無法傳值給數字", 2) end
	if type(obj) == type(noop_func) or type(input_obj) == type(noop_func) then error("無法傳值給函數", 2) end
	if input_obj == nil then error("無法傳值給空值", 2) end 
	input_obj[member] = value
	return value
end

function p._object(obj,...)
	local input_obj = obj
	if type(obj) == type("string") then input_obj = _G[obj] end
	if type(obj) == type(0) then return obj end
	if type(obj) == type(noop_func) then return obj end
	if input_obj == nil then return nil end 
	local members = {...}
	if #members > 0 then
		local it_obj = input_obj
		for i = 1,#members do
			if type(it_obj) ~= type({}) then return nil end
			it_obj = (it_obj or {})[members[i]]
			if it_obj == nil then return nil end
		end
		return it_obj
	end
	return input_obj
end

function p._typeof(obj)
	if type(obj) == type({}) then 
		if obj.numberType then return type(0) end
		local is_array = true
		for index, data in pairs(obj) do
			if not tonumber(index) and index ~= 'metatable' then
				is_array = false
				break
			end
		end
		if is_array then return 'array' end
	end
	return type(obj)
end

function p._array(...)
	return {...}
end

function p._length(obj)
	if type(obj) == type({}) then 
		if obj.numberType then return 1 end
		local max_index = 0
		for key, data in pairs(obj) do
			local index = tonumber(key)
			if (index or 0) > max_index then
				max_index = index
			end
		end
		return max_index
	elseif type(obj) == type("string") then 
		return mw.ustring.len(obj)
	else
		return 1
	end
end

---------------------- 數論相關 ----------------------
function p._findnext(func, x)
	local it = to_number(x) + 1
	if type(func) ~= type(noop_func) then
		if math_lib.abs(to_number(func)) < 1e-14 then
			return to_number("inf")
		else
			return it
		end
	end
	local checker = func(it)
	while math_lib.abs(to_number(checker)) < 1e-14 do
		it = it + 1
		checker = func(it)
	end
	return it
end

function p._findlast(func, x)
	local it = to_number(x) - 1
	if type(func) ~= type(noop_func) then
		if math_lib.abs(to_number(func)) < 1e-14 then
			return to_number("-inf")
		else
			return it
		end
	end
	local checker = func(it)
	while math_lib.abs(to_number(checker)) < 1e-14 do
		it = it - 1
		checker = func(it)
	end
	return it
end
local function key_sort(t)
	if type(t) ~= type({"table"}) then return {t} end
	local key_list = {}
	for k,v in pairs(t) do key_list[#key_list + 1] = k end
	table.sort(key_list)
	return key_list
end
local function get_divisor(n, combination)
	local is_complex = math_lib.abs(math_lib.nonRealPart(n)) > 1e-14
	local factors = {}
	if math_lib.abs(math_lib.floor(n)-n) < 1e-14 then
		local lib_factor = require('Module:Factorization')
		factors = (lib_factor[is_complex and "_gaussianFactorization" or "_factorization"])(
			is_complex and n or tonumber(tostring(n))
		)
	else return combination and {{n}} or {} end
	if not combination then return factors end
	local gened=require('Module:Combination').getCombinationGenerator()
	gened:init(factors,0)
	return gened:findSubset()
end

function p._primedivisor(_n, _x)
	local n = to_number(_n)
	if math_lib.abs(n) < 1e-14 then return 0 end
	
	local is_complex = math_lib.abs(math_lib.nonRealPart(n)) > 1e-14
	if not is_complex then n = math_lib.abs(n) end
	local primedivisors = key_sort(get_divisor(n, false))
	return primedivisors[math_lib.abs(to_number(_x or #primedivisors))] or 0
end

function p._eulerphi(_n, _x)
	local n = to_number(_n)
	if math_lib.abs(n) < 1e-14 then return 0 end
	
	local is_complex = math_lib.abs(math_lib.nonRealPart(n)) > 1e-14
	if not is_complex and math_lib.re(n) < 1e-14 then return 0 end
	local primedivisors = get_divisor(n, false)
	local result = 1
	for p,k in pairs(primedivisors) do
		local p_r = to_number(p)
		local k_r = to_number(k)
		result = result * math_lib.pow(p_r, k_r-1) * (p_r-1)
	end
	return result
end

function p._divisor(_n, _x)
	local n = to_number(_n)
	local function _index(total) return (total <= 2) and total or (total - 1) end
	if math_lib.abs(n) < 1e-14 then return 0 end
	
	local is_complex = math_lib.abs(math_lib.nonRealPart(n)) > 1e-14
	if not is_complex then n = math_lib.abs(n) end
	local combination = get_divisor(n, true)
	local divisors = {}
	for i=1,#combination do
		local divisor = to_number(1)
		for j=1,#(combination[i]) do
			divisor = divisor * to_number(combination[i][j])
		end
		divisors[#divisors+1] = divisor
	end
	table.sort(divisors, function(a,b) return math_lib.abs(a) < math_lib.abs(b) end)
	return divisors[math_lib.abs(to_number(_x or _index(#divisors)))] or 0
end

function p._divisorsigma(_x, _n)
	local x = to_number(1), n
	if _n == nil then 
		n = to_number(_x)
	else
		x = to_number(_x)
		n = to_number(_n)
	end
	if math_lib.abs(n) < 1e-14 then return 0 end

	local is_complex = math_lib.abs(math_lib.nonRealPart(n)) > 1e-14
	if not is_complex then n = math_lib.abs(n) end
	local combination = get_divisor(n, true)
	local sum = to_number(0)
	for i=1,#combination do
		local divisor = to_number(1)
		for j=1,#(combination[i]) do
			divisor = divisor * to_number(combination[i][j])
		end
		sum = sum + math_lib.pow(divisor, x)
	end
	return sum
end

---------------------- 朗伯W函數 ----------------------
local function zexpz(z) return math_lib.exp(z) * z end
--The derivative of z * exp(z) = exp(z) + z * exp(z)
local function zexpz_d(z) return math_lib.exp(z) + math_lib.exp(z) * z end
--The second derivative of z * exp(z) = 2. * exp(z) + z * exp(z)
local function zexpz_dd(z)return math_lib.exp(z) * 2 + math_lib.exp(z) * z end
--Determine the initial point for the root finding
local function LWInitPoint(_z, k)
	local z = to_number(_z)
	local two_pi_k_I = math_lib.i * 2 * math_lib.pi * k
	local ip = math_lib.log(z) + two_pi_k_I - math_lib.log(math_lib.log(z) + two_pi_k_I) --initial point coming from the general asymptotic approximation
	local p = math_lib.sqrt((math_lib.e * z + 1) * 2) --used when we are close to the branch cut around zero and when k=0,-1

	if math_lib.abs(-(-math_lib.exp(-1)) + z) <= 1 then --we are close to the branch cut, the initial point must be chosen carefully
		if k == 0 then ip = -math_lib[1] + p - 1/3 * math_lib.pow(p, 2) + 11/72 * math_lib.pow(p, 3) end
		if k == 1 and math_lib.im(z) < 0 then ip = -math_lib[1] - p - 1/3 * math_lib.pow(p, 2) - 11/72 * math_lib.pow(p, 3) end
		if k == -1 and math_lib.im(z) > 0 then ip = -math_lib[1] - p - 1/3 * math_lib.pow(p, 2) - 11/72 * math_lib.pow(p, 3) end
	end

	if k == 0 and math_lib.abs(z - 0.5) <= 0.5 then ip = ((z * 7.061302897 + 0.1237166) * 0.35173371) / ((z * 2 + 1) * 0.827184 + 2) end-- (1,1) Pade approximant for W(0,a)

	if k == -1 and math_lib.abs(z - 0.5) <= 0.5 then ip = -(((math_lib.i * 4.22096 +
		2.2591588985) * ((-math_lib.i * 33.767687754 - 14.073271) * z - (-math_lib.i * 19.071643 + 
			12.7127) * (z*2 + 1))) / (-(-math_lib.i*10.629721 + 17.23103) * (z*2 + 1) + 2)) end -- (1,1) Pade approximant for W(-1,a)

	return ip;
end

function p._LambertW(_z, _k)
	local z = to_number(_z)
	local k = to_number(_k) or to_number(0)
	local _2 = math_lib[1] * 2
	if math_lib.abs(math_lib.nonRealPart(k)) > 1e-14 then error("朗伯W函数的k只能是實數") end
	k = math_lib.re(k)
	--For some particular z and k W(z,k) has simple value:
	if math_lib.abs(z) == 0 then return (k == 0) and 0 or to_number(-math.huge) end
	if z == -math_lib.exp(-1) and (k == 0 or k == -1) then return -math_lib[1] end
	if z == math_lib.exp(1) and k == 0 then return math_lib[1]+0 end

	--Halley method begins
	local w, wprev = LWInitPoint(z, k), LWInitPoint(z, k) --intermediate values in the Halley method
	local maxiter = 30 --max number of iterations. This eliminates improbable infinite loops
	local iter = 0 --iteration counter
	local prec = 1e-30; --difference threshold between the last two iteration results (or the iter number of iterations is taken)

	wprev = w
	w = w - _2 *((zexpz(w) - z) * zexpz_d(w)) /
		(_2*math_lib.pow(zexpz_d(w),2) - (zexpz(w) - z)*zexpz_dd(w))
	iter = iter + 1
	while ((math_lib.abs(w - wprev) > prec) and iter < maxiter) do
		wprev = w
		w = w - _2 *((zexpz(w) - z) * zexpz_d(w)) /
			(_2*math_lib.pow(zexpz_d(w),2) - (zexpz(w) - z)*zexpz_dd(w))
		iter = iter + 1
	 end
	return w
end

---------------------- 範數 ----------------------
function p._norm(_z, _p)
	local p_value = to_number(_p or 2)
	local check_inf = tostring(_p):match("[Ii][Nn][Ff]")
	local abs_p, re_p = math_lib.abs(p_value), math_lib.re(p_value)
	local value_list = {}
	if type(_z) == type(0) or type(_z) == type("string") or (type(_z) == type({}) and _z.numberType) then
		local z = to_number(_z)
		if type(math_lib.dot) == type(noop_func) then
			if type(math_lib.elements) == type({}) and #(math_lib.elements) > 0 then
				for i=1,#(math_lib.elements) do value_list[#value_list + 1] = math_lib.dot(z, math_lib.elements[i]) end
			else return math_lib.abs(z) end
		else return math_lib.abs(z) end
	elseif type(_z) == type({}) and #_z > 0 then
		for i=1,#_z do value_list[#value_list + 1] = to_number(_z[i]) or to_number(0) end
	end
	if #value_list > 0 then
		local norm_sum, norm_max, norm_min, non_zero_count = 0, -1, tonumber("inf"), 0
		for i=1,#value_list do 
			local abs_value = math_lib.abs(value_list[i])
			if abs_value > norm_max then norm_max = abs_value end
			if abs_value < norm_min then norm_min = abs_value end
			if abs_value ~= 0 then norm_sum = math_lib.pow(abs_value, p_value) + norm_sum end
			if abs_value > 1e-14 then non_zero_count = non_zero_count + 1 end
		end
		return check_inf and (re_p > 0 and norm_max or norm_min) or 
			(abs_p >= 1 and math_lib.pow(norm_sum, math_lib.inverse(p_value)) or 
			(abs_p ~= 0 and norm_sum or non_zero_count))
	end
	error("無效的范數")
end

return p

[1] Wrench, J.W. (1968). Concerning two series for the gamma function. Mathematics of Computation, 22, 617–626. and
Wrench, J.W. (1973). Erratum: Concerning two series for the gamma function. Mathematics of Computation, 27, 681–682.

[2] Viktor T. Toth. "Programmable Calculators: Calculators and the Gamma Function". 2006. （原始内容存档于2007-02-23）.

[3] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds. NIST Digital Library of Mathematical Functions.

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