Kdb+ Lamel

K4 and Kdb+ are trademarks of Kx - Systems

q command	by sample .. last update 22-Dec-2013 18:45
.Q locale(static)	.Q.a(lower alpha),.Q.A(upper alpha),.Q.n (digits),.Q.k (version)
acos	acos -1f .. 3.141593
acot nyi	acot:{acos x%sqrt 1+x*x}
+ [addition]	4h+0xaabb .. 174 191,4h+9j .. 13j
all	all 11111b .. 1b
and	0011b and 1010b .. 0010b
any	any 10001b .. 1b
@ [apply]	"denubolar "@4 6 7 0 1 9 8 3 2 2 1 8 .. "blade runner"
arcosh nyi	{log x+sqrt (x*x)-1f} where x >= 1 (see also under section "COMPLEX OPERATORS")
arcoth nyi	{0.5*log (x+1f)%x-1f} where \|x\| > 1 (see also under section "COMPLEX OPERATORS")
arsinh nyi	{log x+sqrt 1f+x*x} (see also under section "COMPLEX OPERATORS")
artanh nyi	{0.5*log (1f+x)%1f-x} where \|x\| < 1 (see also under section "COMPLEX OPERATORS")
asc	asc "cool" .. `s#"cloo"
asin	(180*asin 0.5)%acos -1f .. 30f
association	("silver"!0x0a114dab762e)@"liver" .. 0x4d11ab762e
atan	(acos -1f) ~4*atan 1 .. 1b (for arccot see `acot)
avg	avg 10 20 30 40 .. 25f
bid/ask notation	"F"$("1 19/64";"2 3/4") .. 1.296875 2.75
bin	7 8 9 10 11 12 13 19 bin 19 9 .. 7 2
cast	99^9 0 0n 0 7 0n 4f .. 9 0 99 0 7 99 4f
@[f;x;msg] [catch]	{@[{x+2};x;`error]} each (4;`4;"m";8j) .. (6;`error;`error;10j) protected file read: {@[read0;x;`unavailable]}@`:F.txt
-3! character representation	-3!(`karl`marx;4 3 2 1;001b) .. "(`karl`marx;4 3 2 1;001b)"
commands	\t .. timer \t f x .. time \P .. precision \p .. port (\p -n reset port) \ .. OS command \v .. variables \d .. dictionary \l .. load script \S .. random seed .. -314159 default \c .. console setting 25 x 80 (default) \C .. browser setting 36 x 2000 (default) \\ .. good bye
$ [condition]	$[1b;`right;`wrong] .. `right
cos	cos (acos -1)%3 .. 0.5
cosh nyi	{0.5*(1%c)+c:exp neg x}
count	count (`aa`bb;`cc`dd) .. 2
cross	`E1`E2`E3 cross `M1`M2 .. (`E1`M1;`E1`M2;`E2`M1;`E2`M2;`E3`M1;`E3`M2)
csv	","
cut	3 cut/:("delta";"airlines") .. (("del";"ta");("air";"lin";"es"))
date arithmetics (m)	2001.01m+key 6 .. 2001.01 2001.02 2001.03 2001.04 2001.05 2001.06m
date arithmetics (m)	2002.03m-2001.01m+key 6 .. 14 13 12 11 10 9
date arithmetics (d)	2006.01.20-til 5 .. 2006.01.20 2006.01.19 2006.01.18 2006.01.17 2006.01.16
date arithmetics (d)	2011.01.20-2006.01.20-til 5 .. 1826 1827 1828 1829 1830
date arithmetics (ts)	2006.01.22T21:06:22.062+1 .. 2006.01.23T21:06:22.062
date arithmetics (ts)	2006.01.22T21:06:23.062+1%86400000 .. 2006.01.22T21:06:23.063
date projections	r:2006.01.22T21:13:36.531 r.date .. 2006.01.22 r.week .. 2006.01.16 r.time .. 21:13:36.531 r.month .. 2006.01m r.minute .. 21:13 r.second .. 21:13:36 r.year .. 2006 r.ss .. 36 r.hh .. 21 r.mm .. 1 r.dd .. 22 r:.z.Z `int$r.time mod 1000 .. reading the milliseconds `int$t.timestamp mod 1000000000 .. 2011.03.15D22:32:22.109921000 (micro-seconds)
date casting (I)	`hh`mm`ss$r.second .. 21 13 36
date casting (II)	`year`month`dd`week$r .. (2006;2006.01m;22;2006.01.16)
date boundaries & date casting (III)	9999.12.31-`date$.z.Z .. (`date$.z.Z)-1000.01.01
date system command \z is 0(=default) or 1	\z 1 .. "D"$("22/09/2009";"31/12/2000";"01/06/2008") .. returns 2009.09.22 2000.12.31 2008.06.01 \z 0 "D"$("2009/09/22";"2000/12/31";"2008/06/01") or "D"$("2009-09-22";"2000-12-31";"2008-06-01") returns .. 2009.09.22 2000.12.31 2008.06.01
[delimiter suppression] by quote intervals	show ("****";",") 0: enlist ("****";",") 0: enlist "Ian,Racz,\"Edmonton,Alberta\",Canada,\"114St,88Ave\",manager" Ian Racz Edmonton,Alberta Canada 114St,88Ave manager
deltas	deltas (til 5) xexp 2 .. 0 1 3 5 7f
denormalizing	flip exec (exec distinct n from T)#n!v by k from T. T:([n:`a`a`a`b`b`c`d;k:`m1`m2`m3`m1`m3`m2`m4] v:9 8 7 6 5 4 3)
dependencies	Tview :: T
desc	desc 1 5 5 7 6 3 3 .. 7 6 5 5 3 3 1 (desc "abcd" .. "dcba")
dictionary	a dictionary maps lists into lists: a:`list1`list2`list3!(`l`e`f`t;4 5 6 7;0101b) we can use the dot notation: a.list2:4 5 6 7 creation of a Kdb+ table: T:flip a
differ	differ 1 5 5 7 6 3 3 .. 1101110b
distinct	distinct "mississippi" .. "misp"
div	integer part of a division .. 19 div' 4 6 7 8 .. 4 3 2 2 j div i .. (j can be 64-bit integer) example: 87726636552j div 873 .. 100488701j
do	do[n;..;..;]
_ [drop]	-4 _ "applegate" .. "apple"
encryption	md5 "birdflu" .. 0x3292dfabfffba27383d72cab3c4413b5
enlist	enlist 6 .. ,6
= [equal]	`cb=`cx`cs`cb`cbb .. 0010b
except	"icewater" except "weird" .. "cat"
exp	exp til 5 .. 1 2.718282 7.389056 20.08554 54.59815
fills	fills "mis is ip i" .. "mississippi"
? [find first]	"Czechoslovakia"?"o" .. 5
first	first "miami" .. "m"
flip	flip (`aa`bb;`cc`dd) .. (`aa`cc;`bb`dd)
floor	floor 2.77 3.99 -3.99 .. 2 3 -4 (integer again)
$ [formats]	[a] IP address to decimal.. "I"$"10.23.23.11" .. 169285387 [b]"Z"$"20060303-10:13:59.099" unix timestamp to q .. 2006.03.03T10:13:59.099 [c]"Ic"$(("34";"88";"299");("mao";"tse";"tung");0x4a4b4c) .. (34 88 299;("mao";"tse";"tung");"JKL") the lets the characters unchanged [d]"ISc"$(("34";"88";"299");("mao";"tse";"tung");0x4a4b4c) .. (34 88 299;`mao`tse`tung;"JKL") [e]"IS"$(("34";"88";"299");("mao";"tse";"tung");0x4a4b4c) .. (34 88 299;`mao`tse`tung;0x4a4b4c) the lets the bytes unchanged [f] bijection of bytes into ASCII:all ("c"$j)="x"$j:key 256 .. 1b. BUT ("c"$j)~"x"$j:key 256 is 0b
getenv	getenv `TEMP
>= [gte]	"o">="moscow" .. 110110b
> [greater than]	"o">"moscow" .. 100100b
hclose	hclose h .. closing file-communication
hcount	hcount `:file .. # of bytes
hdel	hdel file .. erase file
hopen	hopen `:xy/yx.txt .. h "whatever"
iasc [index ascending]	v@iasc v:1 5 5 7 6 3 3 .. 1 3 3 5 5 6 7
~ [identity]	`e ~ /: (5;`e;"nhl 2006";`roland;"e";`e;0x34) .. 0100010b
idesc [index descending]	v@idesc v:1 5 5 7 6 3 3 .. 7 6 5 5 3 3 1
if	if[true;..;..;]
in	"se" in "russia" .. 10b
infinities	inf:(1b,0xff,0Wh,0W,0Wj,0we,0w) # -1 + inf .. (0;254;32766;2147483646;9223372036854775806j;0w;0w) and 0Wd 0Wz 0Wt (infinities related to date/datetime/time)
inter	"birds"inter"parrots" .. "rs"
inv	inv 1f*(4 1 3;-1 1 2;6 7 17) .. (1.5 2 -0.5;14.5 25 -5.5;-6.5 -11 2.5) /set \P 10 eq.system: 15x + 8y = 3 and x - 10y = 95 .. (inv (15 8f;1 -10f)) mmu (3 95f) .. 5 -9f ;x = 5 , y = -9
, [join]	"Czecho","slovakia" .. "Czechoslovakia"
key	key 5 .. 0 1 2 3 4 # key "silver"!0x0a114dab762e .. "silver"
keyed tables	T:([n:`t`a`w;m:7 6 5] a:`A`a`c;b:`ll`jj`dd)
last	last "miami" .. "i"
<[less than]	"miriam" < "m" .. 010110b
like	"panama" like/: ("am";"pan") .. 11b;("trnce";"france") like "[]*" .. 10b
lj	left join
log	log 7 8 2.71828 -3 0 .. 1.94591 2.079442 0.9999993 0n -0w
lower	"GLOBAL LIMIT SYSTEM" .. "global limit system"
lsq [least squares]	1 1.8 1.5 2.5 lsq flip (1 1f;2 1f;3 1f;5 1f) .. 0.3314286 0.7885714
<= [lte]	"o"<"moscow" .. 001001b
ltrim	deleting leading blanks: ltrim " zurich" .. "zurich"
x$m [matrix multiply]	(2 3f;-1 4f)$(9 2f;4 2f) .. (30 10f;7 6f) For (1 x n) . (n x 1) matrices we can take: (c,c:count v)#prd flip v cross v resp. we do: (flip enlist v)$enlist v
mavg	3 mavg 1 3 5 7 9f .. 1 2 3 5 7f
max	max "texas" .. "x"
maxs	maxs 2 4 2 2 8 7 2 7 .. 2 4 4 4 8 8 8 8, "fox"~maxs"fox" .. 1b
mcount	5 mcount "alabama" .. 0 1 2 3 4 5 5 5
med [median]	med 8 7 6 5 6 7 9 .. 7f
min	min 5 -1 8 2 .. -1
mins	mins 67 8 11 4 6 .. 67 8 8 4 4, "volga"~mins"volga" .. 1b
mmax	3 mmax "DEEPTHOUGHT" .. "DEEPTTTUUUT"
mmin	3 mmin 9 8 7 56 17 87 13 .. 9 8 7 7 7 17 13
mmu	.. see matrix multiply
mod	(key 23) mod 11 .. 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 j mod i (j can be 64-bit integer) example: 87726636552j mod 873 .. 579j
msum	3 msum 1 5 7 9 11 .. 1 6 13 21 27
* [multiply]	3*4 5j .. 12 15j
neg	neg (4;9j;7h;0x77;0N;01b) .. (-4;-9j;-7h;-119;0N;0 -1)
next	next "perth" .. "erth "
not	not 1100b .. 0011b, not 1 1 0 0 .. 0011b
null	null 0N 9 8 7 0N .. 10001b
odbc	h:.odbc.open `filedsn`usr`pwd (db2 connect)
or	1100b or 0101b .. 1101b (works also for Int/Long/Short+Bytes)
peach [parallel each]	peach[>;3 13 76 111 223 227 224 701] .. 11111101b
prd	prd 1+til 5 .. 120f
prds	prds -1 2 -3 4 .. -1 -2 6 24
k)s)q) [prefixing]	k) k4 s) jdbc/odbc sql93 .. q) talking to q server
prev	prev key 5 .. 0N 0 1 2 3
protected execution	see under `catch
rand	rand each 6#50 .. 19 39 49 17 2 28
? [random]	5?10 .. 6 5 7 7 7, -5?10 .. 4 3 7 1 6 or 7?" " returns 7 random chars out of "a-z"
rank	rank 11 12 22 12 98 112 .. 0 1 3 2 4 5
ratios	ratios (til 6) xexp 3 .. 0 0w 8 3.375 2.37037 1.953125
raze	raze (1 3;4 5 6 7;1) .. 1 3 4 5 6 7 1
read0	read0 `:text.txt
read1	read1 `:bytes
reciprocal	reciprocal 2 4 8f .. 0.5 0.25 0.125
(n,m)#v [reshape]	2 10#til 20 .. works only for 2-dim, 2 3 4# .. length error
reverse	reverse "flurry" .. "yrrulf"
rotate	4 rotate "techno" .. "notech"
rounding ..	round:{floor 0.5+x}, roundto:{y*floor 0.5+x%y} -> roundto[34.7873 11.3787 38.6767;0.25 0.2 0.125] .. 34.75 11.4 38.625
i [rownumber]	select i,..
rtrim	.. rtrim "geneva " .. "geneva" .. deleting trailing blanks
select	be aware that .. T:select a,b,a from U ..then cols`T is `a`b`a1
select ?[t;a;b;c]	?[M;enlist (>;`b;2);0b;()] .. select from M where b>2 ?[M;((>;`b;2);(like;`c;"r"));0b;()] .. select from M where (b>2) and c like "r" ?[M;((>;`b;2);(like;`c;"r"));0b;(enlist `a)!enlist (sum;`a)] .. select sum a from M where (b>2) and c like "r" ?[M;((>;`b;2);(like;`c;"r"));0b;(`a`c)!((sum;`a);(last;`c))] .. select sum a,last c from M where (b>2) and c like "r" ?[M;enlist (>;`b;2);(enlist `d)!enlist `d;(enlist `a)!enlist (sum;`a)] select sum a by d from M where b>2
.z.s [self referent]	fac2:{$[x<2;1;$[x<3;x;x*.z.s x-2]]} .. double factorial !!n
serialized tables (set+get)	let be - M:([] a:`u`d`x`w; b:9 8 76 1) `:somedir/T set M or .[`:somedir/T;();:;M] saving as T M:get`:somedir/T reading .. we can also use \l somedir/T `:somedir/T insert N .. inserting `:somedir/T insert `a`b!(`I;8777) ..keyed tables..K:([k:`a`b`c]s:`km`lk`ol;v:98 99 88) .[`:somedir/L;();:;K] or `:somedir/L set K for saving .[`:somedir/L;();,;] N or `:somedir/L insert N for inserting
signum	signum 8 0 5 -3j .. 1 0 1 -1
sin	sin (acos -1)%6 .. 0.5
sinh nyi	{0.5*(1%c)-c:exp neg x}
sockets	h:hopen `:host:port (hclose h)
sqrt	sqrt sums 1+2*til 7 .. 1 2 3 4 5 6 7f
ss [string search]	"quantitative easing" ss "ti" .. 4 8
ssr [string replacement]	ssr["sandman";"andm";"ham"] .. "shaman"
string	string (`abc;878;9j;"anc") .. ("abc";"878";,"9";(,"a";,"n";,"c"))
sublist	(2;5) sublist "waterland" .. "terla"
sums	sums -1 2 -3 4 .. -1 1 -2 2
sv	1 100 1000 sv 5 8 9 .. 508009
system	shell execution ..
tables	tables ` .. listing
# [take]	4#9 .. 9 9 9 9, -5#"adam" .. "madam", 4#9 8 7h .. 9 8 7 9h
tan	1f~tan (acos neg 1)%4 .. 1b
tanh nyi	{(d-c)%(d:1f%c)+c:exp neg x}
til	til 5 .. 0 1 2 3 4, til 6h .. 'type, til 6j .. 'type
trim	trim " ghost riders in the sky " .. "ghost riders in the sky" .. deleting leading/trailing blanks(not in-between excess blanks)
type	bool = 1;byte = 4;short = 5;int = 6;long = 7;real = 8;float = 9; char = 10; symb = 11;month = 13; date = 14;datetime = 15;minute = 17;second = 18;time = 19;enum = 20; table = 98;dict = 99;lambda = 100;assignment = 101 (t: ::, type t = 101); projection = 104;composition = 105; f' = 106;f / = 107;f \ = 108;f': = 109;f /: = 110; f \: = 111; dynamic load = 112
uj	union join (self-explaining)
<> [unequal]	"strawberry" <>"r" .. 1101111001b
union	1 3 4 union (3;`a) .. (1;3;4;`a)
upper	upper"global limit system" .. "GLOBAL LIMIT SYSTEM"
value [execute/assoc.algebra]	value "silver"!0x0a114dab762e .. 0x0a114dab762e
vector conditional	?[11010b;76 12 23 11 32;-4 -5 -9 0 3] .. 76 12 -9 11 3
views	views ` .. listing
vs	0x0 vs/:16 32 64 128 255 .. (0x00000010;0x00000020;0x00000040;0x00000080;0x000000ff) 0b vs/: 1 2 4 .. 00000000000000000000000000000001b 00000000000000000000000000000010b 00000000000000000000000000000100b
wavg	2 3 4 wavg 7 8 22 .. 14f
where	where "i"="mississippi" .. 1 4 7 10
while	while[true;..;..;..;]
within	"pqy" within "dw" .. 110b
wsum	4 3 2 1 wsum 1 3 5 7 .. 30f
xbar	select max amts,min amts by 10000 xbar amts from M (select max amts,.., by 10000f xbar .. will result in a type error) i xbar j (where j can be 64-bit integer) .. example: 70000 xbar 3762 87722 144544j .. 0 70000 140000j
xlog	2.71828 10 xlog/: 1e3 1e-7 2.71828 -4.. (6.90776 3;-16.11811 -7;1 0.4342942;0n 0n)
xor nyi	{xwi[x;y] or xwi[y;x]}
xprev	2 xprev 3 4 8 7 .. 0N 0N 3 4, -2 xprev 100 200 300 400 .. 0N 0N 300 400
xwi nyi	{x and not y}
.Q local	.Q.host 1381711022 .. `host174-60.pool8291.interbusiness.it (.Q.host "I"$"85.114.2.6" .. `cobra.obit.ru)
.z local	(h) host,(u) user,(K) version time, (k) version, (a) IP,(f) script,(z) GMT,(Z) timestamp (s) self referent
z1,z2,z = (re,im), {re,im} are real	some COMPLEX OPERATORS (in K4).. if relevant, then always referring to the principal branch - BUT SUBJECT OF A REVIEW (because of the Power Series etc)
π/2 , π , 2π	pih:0.25pi2:2pi:acos -1
cpi(z=i) , cp1(z=1), cpni(z=-i) cpn1(z=-1) , cp2(z=1-i) , cp3(z=-1+i) cp4(z=1+i) , cp5(z=-1-i)	cpi:0 1f;cp1:1 0f;cpni:0 -1f;cpn1:-1 0f;cp2:1 -1f;cp3:-1 1f;cp4:1 1f;cp5:-1 -1f
cp0(z=0) , oo0(z=π/2+0i)) oo1(z=- π/2+0i)	oo1:-1f*oo0:(pih,0f);cp0:0 0f
ep1(z=-0.5+i3^1/2) , ep2(z=-0.5-i3^1/2)	ep2:1 -1ep1:0.5(-1,sqrt 3)
flat (just decorative)	flat:{$[1e-011>abs x;:0f;:x]}
ap (utility)	ap:{0f+x,'cp0}
signum	signum:{(x>0)-x<0} (available under q)
sinh x, x real	sinh:{0.5*(1%c)-c:exp@-x}
cosh x, x real	cosh:{0.5*(1%c)+c:exp@-x}
tanh x, x real	tanh:{(sinh x)%cosh x}
arsinh x, x real	arsinh:{log x+sqrt 1f+x*x}
arcosh x, x ≥ 1, real	arcosh:{log x+sqrt (x*x)-1f}
artanh x, \|x\| < 1,real	artanh:{0.5*log (1+x)%1-x}
cis x = (cos x; sin x), x is real	cis:{(cos x;sin x)} flat@/:/:cis 0.5pikey 5 .. (1 0 -1 0 1f;0 1 0 -1 0f) means cis(0,π/2,π,3π/2,2π) is 1,i,-1,-i,1
absq(z) , squared absolute value f(z) = \|z\|²	absq:{+/x*x} absq (24 3;10 4) is (676 25f), so \|24+10i\|²=676, \|3+4i\|²=25
absz(z) , absolute value f(z) = \|z\|	absz:{sqrt absq x} absz(3 6;4 8) returns 5 10f .. \|3+4i\|=5, \|6+8i\|=10
deg(x), x real ;(degrees)	deg:{x180f%pi} deg asin 0.5sqrt 0 1 2 3 4 returns 0 30 45 60 90f
arc(x), x real ;(arcus)	arc:{x*pi%180f} arc 0 45 90 135 180 360 .. 0 0.7853982 1.570796 2.356194 3.141593 6.283185
arg(z) , argument of z , (-π < arg z ≤ π) φ = arctan y/x	arg:{(1+c1-c:signum y)acos x%absz[x;y]} and in case of a [0,2π)-based definition: arg2:{(pi2l<0f)+l:(1+c1-c:signum y)*acos x%absz(x;y)} arg[1 0 -1 -1 0 -1 1f;0 1 1 0 -1 -1 -1f] .. 0 1.570796 2.356194 3.141593 -1.570796 -2.356194 -0.7853982; φ(1)=0,φ(i)=π/2,φ(-1+i)=3π/4,φ(-1)=π,φ(-i)=-π/2,φ(-1-i)=-3π/4,φ(1-i)=-π/4
arootz(z;n) n positive integer f(z) = \|z\|^1/ne^iφ/n	arootz:{(cis (arg a)%n)\:(+/aa:0f+x) xexp 1f%2fn:last x} arootz((-236 621;-115 -478);3) .. (4 9f;-5 -2f) (-236-115i)^1/3=4-5i ,(621-478i)^1/3=9-2i arootz((9 82368 -47068;40 94024 -1900);2 6 5) .. (5 7 7f;4 1 -5f) (9+40i)^1/2=5+4i ,(-47068+94024i)^1/6=7+i , ,(-47068-1900i)^1/5=7-5i
dymz(z1;z2) , dyadic multiplication(1)	dymz:{(-/mn;+/(m:x@0)\|n:x@1)} dymz((1 4 7 -2;2 -3 -1 5);(3 -2 -4 6;8 1 -3 -7)) .. (-13 -5 -31 23;14 10 -17 44) (1+2i)(3+8i)=-13+23i ,(4-3i)(-2+i)=-5+10i , (7-i)(-4-3i)=-31-17i , (-2+5i)(6-7i) = 23+44i
dymz2(z₀;z) , dyadic multiplication(2)	dymz2:{((-/am);+/(a:x)*\|m:last x)} dymz2(2 5;(3 6 7;-2 5 -4)) .. (16 -13 34;11 40 27) (2+5i)(3-2i)=16+11i ,(2+5i)(6+5i)=-13+40i , (2+5i)(7-4i)=34+27i
sqrz(z) , square f(z) = z²	sqrz:{dymz(x;x)} sqrz(3 -8 5;9 -2 -3) .. (-72 60 16;54 32 -30f)) .. (3+9i)²=-72+54i;(-8-2i)²=60+32i;(5-3i)²=16-30i
cubz(z) , cube f(z) = z³	cubz:{dymz(x;dymz(x;x))} cubz (6 5;4 3) .. (-72 -10;368 198) .. (6+4i)³=-72+368i;(5+3i)³=-10+198i
conj(z) , conjugated f(z) = z^_	conj:{(*x;-last x)} conj(3 8;1 -2)..(3 8;-1 2f); 3+i → 3-i and 8-2i → 8+2i
invz(z) , inverse f(z) = z^-1	invz:{(x1 -1)%\:sum xx} invz(3 6;4 8) .. (0.12 0.06;-0.16 -0.08) (3+4i)^-1 = 0.12-0.16i , (6+8i)^-1 = 0.06-0.08i
expz(z) ; exponencial f(z) = e^z	expz:{{(exp@x)/:cis last x}} expz(0.5 3;0.5 4) .. (1.446889 -13.12878;0.7904391 -15.20078) e^0.5+0.5i=1.446889+0.7904391i , e³⁺⁴ⁱ=-13.12878-15.20078i
sqrtz(z) , square root f(z)= z^1/2	sqrtz:{(-c),'c:aroot(x;2)} sqrtz(54 -520 60;-30 -138 32) .. (-7.608343 -3 -8 7.608343 3 8;1.97152 23 -2 -1.97152 -23 2) (54-30i)^0.5 = ±(-7.608343+1.97152i) , (60+32i)^0.5 = ±(-8+2i) , (-520-138i)^0.5 = ±(-3+23i)
curtz(z) , cubic root f(z)= z^1/3	curtz:{(c,'dymz2(ep1;c)),'dymz2(ep2;c:aroot(x;3))} curtz (-72 -10;368 198) .. (6 5 -6.464102 -5.098076 0.4641016 0.09807621;4 3 3.196152 2.830127 -7.196152 -5.830127) (-72+368i)^1/3 = [6+4i , -6.464102+3.196152i , 0.4641016+(-7.196152)i] (-10+198i)^1/3 = [5+3i , -5.098076+2.830127i , 0.09807621-5.830127i]
prdz(z) ,product f(z) = Π(z)	prdz:{(/absz . x)cis@ +/arg . x} prdz(1 2 4 0.1;3 -5 -3 0.5) .. 30.6 30.8; (1+3i)(2-5i)(4-3i)(0.1+0.5i) = 30.6+30.8i prdz 5#'(4;7) .. 17684 -29113f , (4+7i)⁵ = 17684-29113i (but using "pownz" below is the better deal in this case)
prdz2(z) ,pairwise product f(z) = (z₀,z₁, ... ,z_n) ⊗ (w₀,w₁, ... ,w_n) = (z₀w₀,z₁w₁, ..., z_nw_n)	prdz2:{((x[0]y[0])-x[1]y[1];(x[0]y[1])+x[1]y[0])} prdz2 . ((3 2 6;-2 -1 5);(9 -7 -11;6 5 11)) ... (39 -9 -121;0 17 11); (3-2i,2-i,6+5i) x (9+6i,-7+5i,-11+11i) = (39,-9+17i,-121+11i)
firstpown(z) ,the first n powers of z:(1,z,z²,...,zⁿ)	firstpow:{c:til 1+last x;u:cis@carg . a:first x;u\:v:(absz . a) xexp c} we are looking for the first 7 exponentials of 0.4+0.7i; -> b:firstpow(0.4 0.7;6) 1 , 0.4 + 0.7i , -0.33 + 0.56i , ... ,0.274527 + 0.007336i (= (0.4+0.7i)⁶)
pownz(z;n) power, n ≥ 0 and integer, n vector or scalar f(z) = zⁿ	pownz:{((absz c) xexp e)/:cis (e:last x)arg c:*x} pownz ((4 3 0.5 7 0;7 9 1.5 2 1);1 2 3 4 5) .. (4 -72 -3.25 1241 3.061617e-016;7 54 -2.25 2520 1) (4+7i)¹ = 4+7i,(3+9i)² = -72+54i,(0.5+1.5i)³ = -3.25-2.25i, (7+2i)⁴ = 1241+2520i, i⁵=i pownz((1 -3 -1;1 0 2);4) .. (-4 81 -7f;4.898587e-016 -3.967856e-014 24) (1+i)⁴ = -4 , (-3)⁴ = 81 , (-1+2i)⁴ = -7+24i
rootn(z;n) , n positive scalar integer f(z)=z^1/n	rootn:{,/'((absz c) xexp 1%e)/:/:cis ((arg c:x)+/:pi2*!e)%e:last x} rootn((-72 -10;368 198);3) .. (6 5 -6.464102 -5.098076 0.4641016 0.09807621;4 3 3.196152 2.830127 -7.196152 -5.830127) .. so: (-72+368i)^1/3 = [6+4i , -6.464102+3.196152i , 0.4641016+(-7.196152)i] (-10+198i)^1/3 = [5+3i , -5.098076+2.830127i , 0.09807621-5.830127i]
hornz(coeff); horner schema complex polynomial evaluation	hornz:{y+(-/cx;+/c\|x)} c:3 -4;p:+(6 9 1 0;1 -2 0 -11);last hornz\[*p;1_p] .. -766 -598f; so: p(z)=(6+i)z³+(9-2i)z²+z-11i, then p(3-4i) = -766-598i
fs1(z) .. special functions f(z)=(1+z²)^1/2	fs1:{sqrtz cp1 + sqrz x} fs1 (3 4;-1 -9) .. (-3.147749 -4.020744 3.147749 4.020744;0.9530619 8.953568 -0.9530619 -8.953568) f(3-i)=[-3.147749+0.9530619i,3.147749-0.9530619i];f(4-9i)=[-4.020744+8.953568i,4.020744-8.953568i]
fs2(z) f(z)=(1-z²)^1/2	fs2:{sqrtz cp1 - sqrz x}
fs3(z) f(z)=(z²-1)^1/2	fs3:{sqrtz cpn1 + sqrz x}
fs4(z) f(z)=iz	fs4:{cp3*\|x}
fs5(z) f(z)=-iz	fs5:{cp2*\|x}
fs6(z) f(z)=(i+z)/(i-z)	fs6:{c:dymz(b+x;invz (b:cpi)-x:ap x);c[;&\|/\|/(0w,-0w)=\:x]:cpn1;if[1=#c:-1_'c;c::'c];:c} fs6 (2 0 0;1 1 0w) .. (-1 0n -1;-1 0n 0) f(2+i)=-1-i , f(i) is undefined(singularity) , f(INF) = -1 fs6(-1 -6) .. -0.72 0.04, so f(-1-6i)=-0.72+0.04i
fs7(z) f(z)=(1+z)/(1-z)	fs7:{c:dymz(b+x;invz (b:cp1)-x:ap x);c[;&\|/\|/(0w,-0w)=\:x]:cpn1;if[1=#c:-1_'c;c::'c];:c} fs7(0 1 4f;1 1 -5f) .. (0 -1 -1.176471;1 2 -0.2941176) f(i)=i , f(1+i)=-1+2i , f(4-5i)=-1.176471-0.2941176i
sinz z f(z) = sin z	sinz:{((sin a)cosh b;(cos a:x@0)sinh b:x@1)} sinz (1 0 0.8660254;-2 -1 0.5) .. (3.165779 0 0.8589803;-1.959601 -1.175201 0.3375965) sin(1-2i)=3.165779-1.959601i , sin(-i)=-1.175201i , sin((0.5*3^1/2)+0.5i)=0.8589803+0.3375965i
cosz z f(z) = cos z	cosz:{((cos a)cosh b;-(sin a:x@0)sinh b:x@1)} cosz(2 1 3.3;1 1 -1.7) .. (-0.6421481 0.83373 -2.792904;-1.068607 -0.9888977 -0.417337) cos(2+i)=-0.6421481-1.068607i , cos(1+i)=0.83373-0.9888977i , cos(3.3-1.7i)=-2.792904-0.417337i
tanz z f(z) = tan z	tanz:{dymz (sinz x;invz cosz x)} tanz ((17*pi%8),1,4,(sqrt 3);0 16 0.5 1) .. (0.4142136 2.292611e-014 0.7079078 -0.1126422;0 1 0.8408826 1.288976) tan(17π/8)=-1+2^1/2;tan(1+16i)~i;tan(4-0.25i)=0.7079078+0.8408826i;tan(3^1/2+i) = -0.1126422+1.288976i
cotanz z f(z) = cotg z	cotanz:{dymz (cosz x;invz sinz x)} cotanz (4 2 0 1;0 1 1 -4) .. (0.8636912 -0.1713836 0 0.0006099003;-0 -0.8213298 -1.313035 0.9997206) cotg(4)=0.8636912;cotg(2+i)=-0.1713836-0.8213298i;cotg(i)=-1.313035i;cotg(1-4i) = 0.0006099003+0.9997206i
sinhz z f(z)=sinh z	sinhz:{0f^fs5 sinz fs4 x} sinhz (0 0 0 2 ;6,0,(pih),-11) .. (0 0 0 0.01605139;-0.2794155 0 1 3.762159) sinh(6i)=-0.2794155i , sinh(0)=0 , sinh(iπ/2)=i , sinh(2-11i)=0.01605139+3.762159i
coshz z f(z)=cosh z	coshz:{0f^cosz fs4 x} coshz (0 6.2 -4 0w;(40*pi),7.3,0f,0f) .. (1 129.6126 27.30823 0w;-0 209.5251 -0 0) cosh(40πi)=1 , cosh(6.2+7.3i)=129.6126+209.5251i , cosh(-4)=27.30823 , cosh(INF)=INF
tanhz z f(z)=tanh z	tanhz:{l:300f;c:fs5 tanz fs4 x:ap x;c[;&lb]:cpn1;if[1=#c:-1_'c;c::'c];:c}; tanhz(-3 0 0.25 0w -0w;0 3 -7.1 0 0f ) .. (-0.9950548 0 0.4893675 1 -1;0 -0.1425465 -0.9372601 0 0) tanh(-3)=-0.9950548;tanh(3i)=-0.1425465i;tanh(0.25-7.1i)=0.4893675-0.9372601;tanh(± INF) = ± 1
logz z = log \|z\| + i arg z f(z) = log z	logz:{(0.5*log absq x;arg x)} logz (-1 -10 -1;-1,0,sqrt 3) .. (0.3465736 2.302585 0.6931472;-2.356194 3.141593 2.094395) log(-1-i) = 0.3465736-2.356194i , log(-10) = 2.302585+3.141593i , log(-1+3^1/2i) = 0.6931472+2.094395i
asinz = -i log(iz + (1 - z²)^1/2) f(z) = arcsin z	asinz:{fs5 logz (fs2 x)+(2#x)#'fs4 x} asinz (2 1.732051 0;0 1 1f) .. (1.570796 2.153461 3.141593 1.570796 0.9881321 0;1.316958 -1.358838 -0.8813736 -1.316958 1.358838 0.8813736) arcsin(2)=1.570796 ± 1.316958i ; arcsin(3^1/2+i)=0.9881318+1.358838i,2.153461-1.358838i ; arcsin(i)=0.8813736i,π - 0.8813736i
acosz =-i log(z + i(1 - z²)^1/2) f(z) = arccos z	acosz:{oo0-asinz x} acosz (3 0;4 1) .. (-0.9368125 -1.570796 0.9368125 1.570796;2.305509 0.8813736 -2.305509 -0.8813736) arccos(3+4i)=± (0.9368125-2.305509i), arccos(i)=± (1.570796-0.8813736i)
atanz = 0.5i log [(i + z)/(i - z)] f(z) = arctan z	atanz:{c:0.5fs4 logz fs6 x:ap x;l:last x;c[;&(0w=s:x)\|l in 0w,-0w]:oo0;c[;&-0w=s]:oo1;c[;&(0f=s)&1f< l]:cp3;if[1=#c:-1_'c;c:*:'c];:c} atanz (-5 0 -0w;8 -2 3) .. (-1.514221 -1.570796 -1.570796;0.0898435 -0.5493061 -0) arctan(-5+8i)=-1.514221+0.0898435i, arctan(-2i)=((-π/2)-0.5493061i) , arctan((-INF)+3i) = -π/2
arsinhz = log (z + (z² + 1)^1/2) f(z)=arsinh z	arsinhz:{logz (fs1 x)+(2#x)#'x} arsinhz (-100 0 0;5 0 100 ) .. (5.299591 0 -5.298292 -5.299591 0 5.298292;3.091637 3.141593 1.570796 0.04995591 0 1.570796) arsinh(-100+5i) = ± 5.299591+3.091637i ,arsinh(0) = 0 or πi, arsinh(100i) = ± 5.298292+πi/2
arcoshz = log (z + (z² - 1)^1/2) f(z)=arcosh z	arcoshz:{logz (fs3 x)+(2#x)#'x} arcoshz (-100 0 0;5 0 100 ) .. (5.299541 0 -5.298342 -5.299541 0 5.298342;3.091632 -1.570796 -1.570796 -3.091632 1.570796 1.570796) arcosh(-100+5i) = ±(5.299591+3.091637i) ,arcosh(0) = 0 ± πi/2, arcosh(100i) = ±(5.298292+πi/2)
artanhz = 0.5 log [(1+z)/(1-z)] f(z) = artanh z	artanhz:{c:0.5logz fs7 x:ap x;c[;&(0f<x)&0f=last x]:cp2;if[1=#c:-1_'c;c:*:'c];:c} artanhz (1 0 1 2f;1 0 -3 0f) .. (0.4023595 0 0.0919312 0.5493061;1.017222 0 -1.276795 -1.570796) artanh(1+i) = 0.4023595+1.017222i , artanh(0) = 0 , artanh(1-3i) = 0.0919312-1.276795i , artanh(2) = 0.5493061-1.570796i
complex matrix operations a 1st word	Let be M a complex matrix. We can represent M as Re(M) + iIm(M) (Form 1) or in the more visible form just as M where each element of M is a pair of real numbers (r,c) (Form 2) Form 1 to Form 2: tp12:{x,''y} Form 2 to Form 1: rm:{:''x} (real part) and cm:{last''x} (complex part)
k)cmult (A;B) , A,B of Form 2 complex matrix multiplication	cmult:{a:(rm x;cm x);b:(rm y;cm y);tp12[((r1 mmu r2)-c1 mmu c2);((r1:a@0) mmu c2:b@1)+(c1:a@1) mmu r2:b@0]} example: A:2 2#0f+(2 -1;-3 1;5 3;6 -1) .. ie A[0;0] is 2-i etc. B:2 2#0f+(3 2;9 -1;6 -3;-1 -1) A,B are Form 2 matrices. Then C:cmult[A;B] will return the 2x2 matrix 2 2#(-7 16;21 -9;42 -5;41 17) .. ie C[0;0] is -7+16i etc
k)invc M , M of Form 2 complex matrix inversion	invc:{C:(c:#x)#!:0f+((rm x),'cm x),((-1f*cm x),'rm x);:tp12[C[;i];C[;c+i:!c]]} example: M:2 2#(2 1;-1 9;7 1;-5 -11) .. ie M is a 2 x 2 complex matrix with M[0;0] = 2+i Then L:invc M represents M^-1 .. L is then (rounded to 4 decimals) 2 2#(0.1089 -0.0770;0.0996 -0.0078;-0.0037 -0.0780;-0.0067 0.0238) .. ie L[0;0] is 0.1089-0.077i etc
Powerseries Σ _nc_nzⁿ .. n>-1, integer z, c_j complex numbers	Example 1: Approximate e^0.4+0.7i by a 7-term Powerseries: sum each prdz2 . (flip (1%(*\)1,1_til 7),'0f;firstpow(0.4 0.7;6)) ... 1.140992+0.9610174i Exact value: e^0.4+0.7i = 1.14101..+0.9610599i Example 2:

q command

by sample .. last update 22-Dec-2013 18:45

.Q locale(static)

.Q.a(lower alpha),.Q.A(upper alpha),.Q.n (digits),.Q.k (version)

acos

acos -1f .. 3.141593

acot *nyi*

acot:{acos x%sqrt 1+x*x}

+ [addition]

4h+0xaabb .. 174 191,4h+9j .. 13j

all

all 11111b .. 1b

and

0011b and 1010b .. 0010b

any

any 10001b .. 1b

@ [apply]

"denubolar "@4 6 7 0 1 9 8 3 2 2 1 8 .. "blade runner"

arcosh *nyi*

{log x+sqrt (x*x)-1f} where x >= 1 (see also under section "COMPLEX OPERATORS")

arcoth *nyi*

{0.5*log (x+1f)%x-1f} where |x| > 1 (see also under section "COMPLEX OPERATORS")

arsinh *nyi*

{log x+sqrt 1f+x*x} (see also under section "COMPLEX OPERATORS")

artanh *nyi*

{0.5*log (1f+x)%1f-x} where |x| < 1 (see also under section "COMPLEX OPERATORS")

asc

asc "cool" .. `s#"cloo"

asin

(180*asin 0.5)%acos -1f .. 30f

association

("silver"!0x0a114dab762e)@"liver" .. 0x4d11ab762e

atan

(acos -1f) ~4*atan 1 .. 1b (for arccot see `acot)

avg

avg 10 20 30 40 .. 25f

bid/ask notation

"F"$("1 19/64";"2 3/4") .. 1.296875 2.75

bin

7 8 9 10 11 12 13 19 bin 19 9 .. 7 2

cast

99^9 0 0n 0 7 0n 4f .. 9 0 99 0 7 99 4f

@[f;x;msg] [catch]

{@[{x+2};x;`error]} each (4;`4;"m";8j) .. (6;`error;`error;10j)
protected file read: {@[read0;x;`unavailable]}@`:F.txt

-3! character representation

-3!(`karl`marx;4 3 2 1;001b) .. "(`karl`marx;4 3 2 1;001b)"

commands

\t .. timer
\t f x .. time
\P .. precision
\p .. port (\p -n reset port)
\ .. OS command
\v .. variables
\d .. dictionary
\l .. load script
\S .. random seed .. -314159 default
\c .. console setting 25 x 80 (default)
\C .. browser setting 36 x 2000 (default)
\\ .. good bye

$ [condition]

$[1b;`right;`wrong] .. `right

cos

cos (acos -1)%3 .. 0.5

cosh *nyi*

{0.5*(1%c)+c:exp neg x}

count

count (`aa`bb;`cc`dd) .. 2

cross

`E1`E2`E3 cross `M1`M2 .. (`E1`M1;`E1`M2;`E2`M1;`E2`M2;`E3`M1;`E3`M2)

csv

","

cut

3 cut/:("delta";"airlines") .. (("del";"ta");("air";"lin";"es"))

date arithmetics (m)

2001.01m+key 6 .. 2001.01 2001.02 2001.03 2001.04 2001.05 2001.06m

date arithmetics (m)

2002.03m-2001.01m+key 6 .. 14 13 12 11 10 9

date arithmetics (d)

2006.01.20-til 5 .. 2006.01.20 2006.01.19 2006.01.18 2006.01.17 2006.01.16

date arithmetics (d)

2011.01.20-2006.01.20-til 5 .. 1826 1827 1828 1829 1830

date arithmetics (ts)

2006.01.22T21:06:22.062+1 .. 2006.01.23T21:06:22.062

date arithmetics (ts)

2006.01.22T21:06:23.062+1%86400000 .. 2006.01.22T21:06:23.063

date projections

r:2006.01.22T21:13:36.531
r.date .. 2006.01.22
r.week .. 2006.01.16
r.time .. 21:13:36.531
r.month .. 2006.01m
r.minute .. 21:13
r.second .. 21:13:36
r.year .. 2006
r.ss .. 36
r.hh .. 21
r.mm .. 1
r.dd .. 22

r:.z.Z
`int$r.time mod 1000 .. reading the milliseconds
`int$t.timestamp mod 1000000000 .. 2011.03.15D22:32:22.109921000 (micro-seconds)

date casting (I)

`hh`mm`ss$r.second .. 21 13 36

date casting (II)

`year`month`dd`week$r .. (2006;2006.01m;22;2006.01.16)

date boundaries &
date casting (III)

9999.12.31-`date$.z.Z .. (`date$.z.Z)-1000.01.01

date system command
\z is 0(=default) or 1

\z 1 .. "D"$("22/09/2009";"31/12/2000";"01/06/2008") .. returns 2009.09.22 2000.12.31 2008.06.01
\z 0 "D"$("2009/09/22";"2000/12/31";"2008/06/01") or "D"$("2009-09-22";"2000-12-31";"2008-06-01") returns ..
2009.09.22 2000.12.31 2008.06.01

[delimiter suppression]
by quote intervals

show ("******";",") 0: enlist ("******";",") 0: enlist "Ian,Racz,\"Edmonton,Alberta\",Canada,\"114St,88Ave\",manager"
Ian
Racz
Edmonton,Alberta
Canada
114St,88Ave
manager

deltas

deltas (til 5) xexp 2 .. 0 1 3 5 7f

denormalizing

flip exec (exec distinct n from T)#n!v by k from T.
T:([n:`a`a`a`b`b`c`d;k:`m1`m2`m3`m1`m3`m2`m4] v:9 8 7 6 5 4 3)

dependencies

Tview :: T

desc

desc 1 5 5 7 6 3 3 .. 7 6 5 5 3 3 1 (desc "abcd" .. "dcba")

dictionary

a dictionary maps lists into lists:
a:`list1`list2`list3!(`l`e`f`t;4 5 6 7;0101b)
we can use the dot notation: a.list2:4 5 6 7
creation of a Kdb+ table: T:flip a

differ

differ 1 5 5 7 6 3 3 .. 1101110b

distinct

distinct "mississippi" .. "misp"

div

integer part of a division .. 19 div' 4 6 7 8 .. 4 3 2 2
j div i .. (j can be 64-bit integer) example: 87726636552j div 873 .. 100488701j

do[n;..;..;]

_ [drop]

-4 _ "applegate" .. "apple"

encryption

md5 "birdflu" .. 0x3292dfabfffba27383d72cab3c4413b5

enlist

enlist 6 .. ,6

= [equal]

`cb=`cx`cs`cb`cbb .. 0010b

except

"icewater" except "weird" .. "cat"

exp

exp til 5 .. 1 2.718282 7.389056 20.08554 54.59815

fills

fills "mis is ip i" .. "mississippi"

? [find first]

"Czechoslovakia"?"o" .. 5

first

first "miami" .. "m"

flip

flip (`aa`bb;`cc`dd) .. (`aa`cc;`bb`dd)

floor

floor 2.77 3.99 -3.99 .. 2 3 -4 (integer again)

$ [formats]

[a] IP address to decimal.. "I"$"10.23.23.11" .. 169285387
[b]"Z"$"20060303-10:13:59.099" unix timestamp to q .. 2006.03.03T10:13:59.099
[c]"I*c"$(("34";"88";"299");("mao";"tse";"tung");0x4a4b4c) .. (34 88 299;("mao";"tse";"tung");"JKL")
the * lets the characters unchanged
[d]"ISc"$(("34";"88";"299");("mao";"tse";"tung");0x4a4b4c) .. (34 88 299;`mao`tse`tung;"JKL") [e]"IS*"$(("34";"88";"299");("mao";"tse";"tung");0x4a4b4c) .. (34 88 299;`mao`tse`tung;0x4a4b4c)
the * lets the bytes unchanged
[f] bijection of bytes into ASCII:all ("c"$j)="x"$j:key 256 .. 1b. BUT ("c"$j)~"x"$j:key 256 is 0b

getenv

getenv `TEMP

>= [gte]

"o">="moscow" .. 110110b

> [greater than]

"o">"moscow" .. 100100b

hclose

hclose h .. closing file-communication

hcount

hcount `:file .. # of bytes

hdel

hdel file .. erase file

hopen

hopen `:xy/yx.txt .. h "whatever"

iasc [index ascending]

v@iasc v:1 5 5 7 6 3 3 .. 1 3 3 5 5 6 7

~ [identity]

`e ~ /: (5;`e;"nhl 2006";`roland;"e";`e;0x34) .. 0100010b

idesc [index descending]

v@idesc v:1 5 5 7 6 3 3 .. 7 6 5 5 3 3 1

if[true;..;..;]

"se" in "russia" .. 10b

infinities

inf:(1b,0xff,0Wh,0W,0Wj,0we,0w) # -1 + inf .. (0;254;32766;2147483646;9223372036854775806j;0w;0w)
and 0Wd 0Wz 0Wt (infinities related to date/datetime/time)

inter

"birds"inter"parrots" .. "rs"

inv

inv 1f*(4 1 3;-1 1 2;6 7 17) .. (1.5 2 -0.5;14.5 25 -5.5;-6.5 -11 2.5) /set \P 10
eq.system: 15x + 8y = 3 and x - 10y = 95 .. (inv (15 8f;1 -10f)) mmu (3 95f) .. 5 -9f ;x = 5 , y = -9

, [join]

"Czecho","slovakia" .. "Czechoslovakia"

key

key 5 .. 0 1 2 3 4 # key "silver"!0x0a114dab762e .. "silver"

keyed tables

T:([n:`t`a`w;m:7 6 5] a:`A`a`c;b:`ll`jj`dd)

last

last "miami" .. "i"

<[less than]

"miriam" < "m" .. 010110b

"panama" like/: ("*am*";"pan*") .. 11b;("tr*nce";"france") like "*[*]*" .. 10b

left join

log

log 7 8 2.71828 -3 0 .. 1.94591 2.079442 0.9999993 0n -0w

lower

"GLOBAL LIMIT SYSTEM" .. "global limit system"

lsq [least squares]

1 1.8 1.5 2.5 lsq flip (1 1f;2 1f;3 1f;5 1f) .. 0.3314286 0.7885714

<= [lte]

"o"<"moscow" .. 001001b

ltrim

deleting leading blanks: ltrim " zurich" .. "zurich"

x$m [matrix multiply]

(2 3f;-1 4f)$(9 2f;4 2f) .. (30 10f;7 6f)
For (1 x n) . (n x 1) matrices we can take: (c,c:count v)#prd flip v cross v
resp. we do: (flip enlist v)$enlist v

mavg

3 mavg 1 3 5 7 9f .. 1 2 3 5 7f

max

max "texas" .. "x"

maxs

maxs 2 4 2 2 8 7 2 7 .. 2 4 4 4 8 8 8 8, "fox"~maxs"fox" .. 1b

mcount

5 mcount "alabama" .. 0 1 2 3 4 5 5 5

med [median]

med 8 7 6 5 6 7 9 .. 7f

min

min 5 -1 8 2 .. -1

mins

mins 67 8 11 4 6 .. 67 8 8 4 4, "volga"~mins"volga" .. 1b

mmax

3 mmax "DEEPTHOUGHT" .. "DEEPTTTUUUT"

mmin

3 mmin 9 8 7 56 17 87 13 .. 9 8 7 7 7 17 13

mmu

.. see matrix multiply

mod

(key 23) mod 11 .. 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0
j mod i (j can be 64-bit integer) example: 87726636552j mod 873 .. 579j

msum

3 msum 1 5 7 9 11 .. 1 6 13 21 27

* [multiply]

3*4 5j .. 12 15j

neg

neg (4;9j;7h;0x77;0N;01b) .. (-4;-9j;-7h;-119;0N;0 -1)

next "perth" .. "erth "

not

not 1100b .. 0011b, not 1 1 0 0 .. 0011b

null

null 0N 9 8 7 0N .. 10001b

odbc

h:.odbc.open `filedsn`usr`pwd (db2 connect)

1100b or 0101b .. 1101b (works also for Int/Long/Short+Bytes)

peach [parallel each]

peach[>;3 13 76 111 223 227 224 701] .. 11111101b

prd

prd 1+til 5 .. 120f

prds

prds -1 2 -3 4 .. -1 -2 6 24

k)s)q) [prefixing]

k) k4 s) jdbc/odbc sql93 .. q) talking to q server

prev key 5 .. 0N 0 1 2 3

protected execution

see under `catch

rand

rand each 6#50 .. 19 39 49 17 2 28

? [random]

5?10 .. 6 5 7 7 7, -5?10 .. 4 3 7 1 6 or 7?" " returns 7 random chars out of "a-z"

rank

rank 11 12 22 12 98 112 .. 0 1 3 2 4 5

ratios

ratios (til 6) xexp 3 .. 0 0w 8 3.375 2.37037 1.953125

raze

raze (1 3;4 5 6 7;1) .. 1 3 4 5 6 7 1

read0 `:text.txt

read1 `:bytes

reciprocal

reciprocal 2 4 8f .. 0.5 0.25 0.125

(n,m)#v [reshape]

2 10#til 20 .. works only for 2-dim, 2 3 4# .. length error

reverse

reverse "flurry" .. "yrrulf"

rotate

4 rotate "techno" .. "notech"

rounding ..

round:{floor 0.5+x}, roundto:{y*floor 0.5+x%y} -> roundto[34.7873 11.3787 38.6767;0.25 0.2 0.125] .. 34.75 11.4 38.625

i [rownumber]

select i,..

rtrim

.. rtrim "geneva " .. "geneva" .. deleting trailing blanks

select

be aware that .. T:select a,b,a from U ..then cols`T is `a`b`a1

select ?[t;a;b;c]

?[M;enlist (>;`b;2);0b;()] ..
select from M where b>2

?[M;((>;`b;2);(like;`c;"*r*"));0b;()] ..
select from M where (b>2) and c like "*r*"

?[M;((>;`b;2);(like;`c;"*r*"));0b;(enlist `a)!enlist (sum;`a)] ..
select sum a from M where (b>2) and c like "*r*"

?[M;((>;`b;2);(like;`c;"*r*"));0b;(`a`c)!((sum;`a);(last;`c))] ..
select sum a,last c from M where (b>2) and c like "*r*"

?[M;enlist (>;`b;2);(enlist `d)!enlist `d;(enlist `a)!enlist (sum;`a)]
select sum a by d from M where b>2

.z.s [self referent]

fac2:{$[x<2;1;$[x<3;x;x*.z.s x-2]]} .. double factorial !!n

serialized tables (set+get)

let be - M:([] a:`u`d`x`w; b:9 8 76 1)
`:somedir/T set M or .[`:somedir/T;();:;M] saving as T
M:get`:somedir/T reading .. we can also use \l somedir/T
`:somedir/T insert N .. inserting
`:somedir/T insert `a`b!(`I;8777)

..keyed tables..K:([k:`a`b`c]s:`km`lk`ol;v:98 99 88)

.[`:somedir/L;();:;K] or `:somedir/L set K for saving
.[`:somedir/L;();,;] N or `:somedir/L insert N for inserting

signum

signum 8 0 5 -3j .. 1 0 1 -1

sin

sin (acos -1)%6 .. 0.5

sinh *nyi*

{0.5*(1%c)-c:exp neg x}

sockets

h:hopen `:host:port (hclose h)

sqrt

sqrt sums 1+2*til 7 .. 1 2 3 4 5 6 7f

ss [string search]

"quantitative easing" ss "ti" .. 4 8

ssr [string replacement]

ssr["sandman";"andm";"ham"] .. "shaman"

string

string (`abc;878;9j;"anc") .. ("abc";"878";,"9";(,"a";,"n";,"c"))

sublist

(2;5) sublist "waterland" .. "terla"

sums

sums -1 2 -3 4 .. -1 1 -2 2

1 100 1000 sv 5 8 9 .. 508009

system

shell execution ..

tables

tables ` .. listing

# [take]

4#9 .. 9 9 9 9, -5#"adam" .. "madam", 4#9 8 7h .. 9 8 7 9h

tan

1f~tan (acos neg 1)%4 .. 1b

tanh *nyi*

{(d-c)%(d:1f%c)+c:exp neg x}

til

til 5 .. 0 1 2 3 4, til 6h .. 'type, til 6j .. 'type

trim

trim " ghost riders in the sky " .. "ghost riders in the sky" .. deleting leading/trailing blanks(not in-between excess blanks)

type

bool = 1;byte = 4;short = 5;int = 6;long = 7;real = 8;float = 9; char = 10; symb = 11;month = 13;
date = 14;datetime = 15;minute = 17;second = 18;time = 19;enum = 20;
table = 98;dict = 99;lambda = 100;assignment = 101 (t: ::, type t = 101); projection = 104;composition = 105;
f' = 106;f / = 107;f \ = 108;f': = 109;f /: = 110; f \: = 111; dynamic load = 112

union join (self-explaining)

<> [unequal]

"strawberry" <>"r" .. 1101111001b

union

1 3 4 union (3;`a) .. (1;3;4;`a)

upper

upper"global limit system" .. "GLOBAL LIMIT SYSTEM"

value [execute/assoc.algebra]

value "silver"!0x0a114dab762e .. 0x0a114dab762e

vector conditional

?[11010b;76 12 23 11 32;-4 -5 -9 0 3] .. 76 12 -9 11 3

views

views ` .. listing

0x0 vs/:16 32 64 128 255 .. (0x00000010;0x00000020;0x00000040;0x00000080;0x000000ff)
0b vs/: 1 2 4 ..

00000000000000000000000000000001b
00000000000000000000000000000010b
00000000000000000000000000000100b

wavg

2 3 4 wavg 7 8 22 .. 14f

where

where "i"="mississippi" .. 1 4 7 10

while

while[true;..;..;..;]

within

"pqy" within "dw" .. 110b

wsum

4 3 2 1 wsum 1 3 5 7 .. 30f

xbar

select max amts,min amts by 10000 xbar amts from M
(select max amts,.., by 10000f xbar .. will result in a type error)
i xbar j (where j can be 64-bit integer) .. example: 70000 xbar 3762 87722 144544j .. 0 70000 140000j

xlog

2.71828 10 xlog/: 1e3 1e-7 2.71828 -4.. (6.90776 3;-16.11811 -7;1 0.4342942;0n 0n)

xor *nyi*

{xwi[x;y] or xwi[y;x]}

xprev

2 xprev 3 4 8 7 .. 0N 0N 3 4, -2 xprev 100 200 300 400 .. 0N 0N 300 400

xwi *nyi*

{x and not y}

.Q local

.Q.host 1381711022 .. `host174-60.pool8291.interbusiness.it (.Q.host "I"$"85.114.2.6" .. `cobra.obit.ru)

.z local

(h) host,(u) user,(K) version time, (k) version, (a) IP,(f) script,(z) GMT,(Z) timestamp (s) self referent

z1,z2,z = (re,im), {re,im} are real

some COMPLEX OPERATORS (in K4).. if relevant, then always referring to the principal branch - BUT SUBJECT OF A REVIEW (because of the Power Series etc)

π/2 , π , 2π

pih:0.25*pi2:2*pi:acos -1

cpi(z=i) , cp1(z=1), cpni(z=-i)
cpn1(z=-1) , cp2(z=1-i) , cp3(z=-1+i)
cp4(z=1+i) , cp5(z=-1-i)

cpi:0 1f;cp1:1 0f;cpni:0 -1f;cpn1:-1 0f;cp2:1 -1f;cp3:-1 1f;cp4:1 1f;cp5:-1 -1f

cp0(z=0) , oo0(z=π/2+0i))
oo1(z=- π/2+0i)

oo1:-1f*oo0:(pih,0f);cp0:0 0f

ep1(z=-0.5+i3^1/2) , ep2(z=-0.5-i3^1/2)

ep2:1 -1*ep1:0.5*(-1,sqrt 3)

flat (just decorative)

flat:{$[1e-011>abs x;:0f;:x]}

ap (utility)

ap:{0f+x,'cp0}

signum

signum:{(x>0)-x<0} (available under q)

sinh x, x real

sinh:{0.5*(1%c)-c:exp@-x}

cosh x, x real

cosh:{0.5*(1%c)+c:exp@-x}

tanh x, x real

tanh:{(sinh x)%cosh x}

arsinh x, x real

arsinh:{log x+sqrt 1f+x*x}

arcosh x, x ≥ 1, real

arcosh:{log x+sqrt (x*x)-1f}

artanh x, |x| < 1,real

artanh:{0.5*log (1+x)%1-x}

cis x = (cos x; sin x), x is real

cis:{(cos x;sin x)}

flat@/:/:cis 0.5*pi*key 5 .. (1 0 -1 0 1f;0 1 0 -1 0f)
means cis(0,π/2,π,3π/2,2π) is 1,i,-1,-i,1

absq(z) , squared absolute value

f(z) = |z|²

absq:{+/x*x}

absq (24 3;10 4) is (676 25f), so

|24+10i|²=676, |3+4i|²=25

absz(z) , absolute value

f(z) = |z|

absz:{sqrt absq x}

absz(3 6;4 8) returns 5 10f ..

|3+4i|=5, |6+8i|=10

deg(x), x real ;(degrees)

deg:{x*180f%pi}

deg asin 0.5*sqrt 0 1 2 3 4 returns 0 30 45 60 90f

arc(x), x real ;(arcus)

arc:{x*pi%180f}

arc 0 45 90 135 180 360 .. 0 0.7853982 1.570796 2.356194 3.141593 6.283185

arg(z) , argument of z ,
(-π < arg z ≤ π)

φ = arctan y/x

arg:{(1+c*1-c:signum y)*acos x%absz[x;y]} and in case of a [0,2π)-based definition:
arg2:{(pi2*l<0f)+l:(1+c*1-c:signum y)*acos x%absz(x;y)}

arg[1 0 -1 -1 0 -1 1f;0 1 1 0 -1 -1 -1f] .. 0 1.570796 2.356194 3.141593 -1.570796 -2.356194 -0.7853982;

φ(1)=0,φ(i)=π/2,φ(-1+i)=3π/4,φ(-1)=π,φ(-i)=-π/2,φ(-1-i)=-3π/4,φ(1-i)=-π/4

arootz(z;n) n positive integer

f(z) = |z|^1/ne^iφ/n

arootz:{(cis (arg a)%n)*\:(+/a*a:0f+*x) xexp 1f%2f*n:last x}

arootz((-236 621;-115 -478);3) .. (4 9f;-5 -2f)

(-236-115i)^1/3=4-5i ,(621-478i)^1/3=9-2i

arootz((9 82368 -47068;40 94024 -1900);2 6 5) .. (5 7 7f;4 1 -5f)

(9+40i)^1/2=5+4i ,(-47068+94024i)^1/6=7+i , ,(-47068-1900i)^1/5=7-5i

dymz(z1;z2) ,

dyadic multiplication(1)

dymz:{(-/m*n;+/(m:x@0)*|n:x@1)}

dymz((1 4 7 -2;2 -3 -1 5);(3 -2 -4 6;8 1 -3 -7)) .. (-13 -5 -31 23;14 10 -17 44)

(1+2i)(3+8i)=-13+23i ,(4-3i)(-2+i)=-5+10i , (7-i)(-4-3i)=-31-17i , (-2+5i)(6-7i) = 23+44i

dymz2(z₀;z) ,

dyadic multiplication(2)

dymz2:{((-/a*m);+/(a:*x)*|m:last x)}

dymz2(2 5;(3 6 7;-2 5 -4)) .. (16 -13 34;11 40 27)

(2+5i)(3-2i)=16+11i ,(2+5i)(6+5i)=-13+40i , (2+5i)(7-4i)=34+27i

sqrz(z) , square

f(z) = z²

sqrz:{dymz(x;x)}

sqrz(3 -8 5;9 -2 -3) .. (-72 60 16;54 32 -30f)) ..

(3+9i)²=-72+54i;(-8-2i)²=60+32i;(5-3i)²=16-30i

cubz(z) , cube

f(z) = z³

cubz:{dymz(x;dymz(x;x))}

cubz (6 5;4 3) .. (-72 -10;368 198) ..

(6+4i)³=-72+368i;(5+3i)³=-10+198i

conj(z) , conjugated

f(z) = z^_

conj:{(*x;-last x)}

conj(3 8;1 -2)..(3 8;-1 2f);

3+i → 3-i and 8-2i → 8+2i

invz(z) , inverse

f(z) = z^-1

invz:{(x*1 -1)%\:sum x*x}

invz(3 6;4 8) .. (0.12 0.06;-0.16 -0.08)

(3+4i)^-1 = 0.12-0.16i , (6+8i)^-1 = 0.06-0.08i

expz(z) ; exponencial

f(z) = e^z

expz:{{(exp@*x)*/:cis last x}}

expz(0.5 3;0.5 4) .. (1.446889 -13.12878;0.7904391 -15.20078)

e^0.5+0.5i=1.446889+0.7904391i , e³⁺⁴ⁱ=-13.12878-15.20078i

sqrtz(z) , square root

f(z)= z^1/2

sqrtz:{(-c),'c:aroot(x;2)}

sqrtz(54 -520 60;-30 -138 32) .. (-7.608343 -3 -8 7.608343 3 8;1.97152 23 -2 -1.97152 -23 2)

(54-30i)^0.5 = ±(-7.608343+1.97152i) , (60+32i)^0.5 = ±(-8+2i) , (-520-138i)^0.5 = ±(-3+23i)

curtz(z) , cubic root

f(z)= z^1/3

curtz:{(c,'dymz2(ep1;c)),'dymz2(ep2;c:aroot(x;3))}

curtz (-72 -10;368 198) ..
(6 5 -6.464102 -5.098076 0.4641016 0.09807621;4 3 3.196152 2.830127 -7.196152 -5.830127)

(-72+368i)^1/3 = [6+4i , -6.464102+3.196152i , 0.4641016+(-7.196152)i]
(-10+198i)^1/3 = [5+3i , -5.098076+2.830127i , 0.09807621-5.830127i]

prdz(z) ,product

f(z) = Π(z)

prdz:{(*/absz . x)*cis@ +/arg . x}

prdz(1 2 4 0.1;3 -5 -3 0.5) .. 30.6 30.8;

(1+3i)(2-5i)(4-3i)(0.1+0.5i) = 30.6+30.8i

prdz 5#'(4;7) .. 17684 -29113f ,

(4+7i)⁵ = 17684-29113i (but using "pownz" below is the better deal in this case)

prdz2(z) ,pairwise product

f(z) = (z₀,z₁, ... ,z_n) ⊗ (w₀,w₁, ... ,w_n) = (z₀w₀,z₁w₁, ..., z_nw_n)

prdz2:{((x[0]*y[0])-x[1]*y[1];(x[0]*y[1])+x[1]*y[0])}

prdz2 . ((3 2 6;-2 -1 5);(9 -7 -11;6 5 11)) ... (39 -9 -121;0 17 11);

(3-2i,2-i,6+5i) x (9+6i,-7+5i,-11+11i) = (39,-9+17i,-121+11i)

firstpown(z) ,the first n powers
of z:(1,z,z²,...,zⁿ)

firstpow:{c:til 1+last x;u:cis@c*arg . a:first x;u*\:v:(absz . a) xexp c}

we are looking for the first 7 exponentials of 0.4+0.7i;

-> b:firstpow(0.4 0.7;6)

1 , 0.4 + 0.7i , -0.33 + 0.56i , ... ,0.274527 + 0.007336i (= (0.4+0.7i)⁶)

pownz(z;n) power, n ≥ 0 and integer, n vector or scalar

f(z) = zⁿ

pownz:{((absz c) xexp e)*/:cis (e:last x)*arg c:*x}

pownz ((4 3 0.5 7 0;7 9 1.5 2 1);1 2 3 4 5) .. (4 -72 -3.25 1241 3.061617e-016;7 54 -2.25 2520 1)

(4+7i)¹ = 4+7i,(3+9i)² = -72+54i,(0.5+1.5i)³ = -3.25-2.25i, (7+2i)⁴ = 1241+2520i, i⁵=i

pownz((1 -3 -1;1 0 2);4) .. (-4 81 -7f;4.898587e-016 -3.967856e-014 24)

(1+i)⁴ = -4 , (-3)⁴ = 81 , (-1+2i)⁴ = -7+24i

rootn(z;n) , n positive scalar integer

f(z)=z^1/n

rootn:{,/'((absz c) xexp 1%e)*/:/:cis ((arg c:*x)+/:pi2*!e)%e:last x}

rootn((-72 -10;368 198);3) ..
(6 5 -6.464102 -5.098076 0.4641016 0.09807621;4 3 3.196152 2.830127 -7.196152 -5.830127) .. so:

(-72+368i)^1/3 = [6+4i , -6.464102+3.196152i , 0.4641016+(-7.196152)i]
(-10+198i)^1/3 = [5+3i , -5.098076+2.830127i , 0.09807621-5.830127i]

hornz(coeff); horner schema

complex polynomial evaluation

hornz:{y+(-/c*x;+/c*|x)}

c:3 -4;p:+(6 9 1 0;1 -2 0 -11);last hornz\[*p;1_p] .. -766 -598f; so:

p(z)=(6+i)z³+(9-2i)z²+z-11i, then p(3-4i) = -766-598i

fs1(z) .. special functions

f(z)=(1+z²)^1/2

fs1:{sqrtz cp1 + sqrz x}

fs1 (3 4;-1 -9) .. (-3.147749 -4.020744 3.147749 4.020744;0.9530619 8.953568 -0.9530619 -8.953568)

f(3-i)=[-3.147749+0.9530619i,3.147749-0.9530619i];f(4-9i)=[-4.020744+8.953568i,4.020744-8.953568i]

fs2(z)

f(z)=(1-z²)^1/2

fs2:{sqrtz cp1 - sqrz x}

fs3(z)

f(z)=(z²-1)^1/2

fs3:{sqrtz cpn1 + sqrz x}

fs4(z)

f(z)=iz

fs4:{cp3*|x}

fs5(z)

f(z)=-iz

fs5:{cp2*|x}

fs6(z)

f(z)=(i+z)/(i-z)

fs6:{c:dymz(b+x;invz (b:cpi)-x:ap x);c[;&|/|/(0w,-0w)=\:x]:cpn1;if[1=#*c:-1_'c;c:*:'c];:c}

fs6 (2 0 0;1 1 0w) .. (-1 0n -1;-1 0n 0)

f(2+i)=-1-i , f(i) is undefined(singularity) , f(INF) = -1

fs6(-1 -6) .. -0.72 0.04, so

f(-1-6i)=-0.72+0.04i

fs7(z)

f(z)=(1+z)/(1-z)

fs7:{c:dymz(b+x;invz (b:cp1)-x:ap x);c[;&|/|/(0w,-0w)=\:x]:cpn1;if[1=#*c:-1_'c;c:*:'c];:c}

fs7(0 1 4f;1 1 -5f) .. (0 -1 -1.176471;1 2 -0.2941176)

f(i)=i , f(1+i)=-1+2i , f(4-5i)=-1.176471-0.2941176i

sinz z

f(z) = sin z

sinz:{((sin a)*cosh b;(cos a:x@0)*sinh b:x@1)}

sinz (1 0 0.8660254;-2 -1 0.5) .. (3.165779 0 0.8589803;-1.959601 -1.175201 0.3375965)

sin(1-2i)=3.165779-1.959601i , sin(-i)=-1.175201i , sin((0.5*3^1/2)+0.5i)=0.8589803+0.3375965i

cosz z

f(z) = cos z

cosz:{((cos a)*cosh b;-(sin a:x@0)*sinh b:x@1)}

cosz(2 1 3.3;1 1 -1.7) .. (-0.6421481 0.83373 -2.792904;-1.068607 -0.9888977 -0.417337)

cos(2+i)=-0.6421481-1.068607i , cos(1+i)=0.83373-0.9888977i , cos(3.3-1.7i)=-2.792904-0.417337i

tanz z

f(z) = tan z

tanz:{dymz (sinz x;invz cosz x)}

tanz ((17*pi%8),1,4,(sqrt 3);0 16 0.5 1) .. (0.4142136 2.292611e-014 0.7079078 -0.1126422;0 1 0.8408826 1.288976)

tan(17π/8)=-1+2^1/2;tan(1+16i)~i;tan(4-0.25i)=0.7079078+0.8408826i;tan(3^1/2+i) = -0.1126422+1.288976i

cotanz z

f(z) = cotg z

cotanz:{dymz (cosz x;invz sinz x)}

cotanz (4 2 0 1;0 1 1 -4) .. (0.8636912 -0.1713836 0 0.0006099003;-0 -0.8213298 -1.313035 0.9997206)

cotg(4)=0.8636912;cotg(2+i)=-0.1713836-0.8213298i;cotg(i)=-1.313035i;cotg(1-4i) = 0.0006099003+0.9997206i

sinhz z

f(z)=sinh z

sinhz:{0f^fs5 sinz fs4 x}

sinhz (0 0 0 2 ;6,0,(pih),-11) .. (0 0 0 0.01605139;-0.2794155 0 1 3.762159)

sinh(6i)=-0.2794155i , sinh(0)=0 , sinh(iπ/2)=i , sinh(2-11i)=0.01605139+3.762159i

coshz z

f(z)=cosh z

coshz:{0f^cosz fs4 x}

coshz (0 6.2 -4 0w;(40*pi),7.3,0f,0f) .. (1 129.6126 27.30823 0w;-0 209.5251 -0 0)

cosh(40πi)=1 , cosh(6.2+7.3i)=129.6126+209.5251i , cosh(-4)=27.30823 , cosh(INF)=INF

tanhz z

f(z)=tanh z

tanhz:{l:300f;c:fs5 tanz fs4 x:ap x;c[;&lb]:cpn1;if[1=#*c:-1_'c;c:*:'c];:c};

tanhz(-3 0 0.25 0w -0w;0 3 -7.1 0 0f ) .. (-0.9950548 0 0.4893675 1 -1;0 -0.1425465 -0.9372601 0 0)

tanh(-3)=-0.9950548;tanh(3i)=-0.1425465i;tanh(0.25-7.1i)=0.4893675-0.9372601;tanh(± INF) = ± 1

logz z = log |z| + i arg z

f(z) = log z

logz:{(0.5*log absq x;arg x)}

logz (-1 -10 -1;-1,0,sqrt 3) .. (0.3465736 2.302585 0.6931472;-2.356194 3.141593 2.094395)

log(-1-i) = 0.3465736-2.356194i , log(-10) = 2.302585+3.141593i , log(-1+3^1/2i) = 0.6931472+2.094395i

asinz = -i log(iz + (1 - z²)^1/2)

f(z) = arcsin z

asinz:{fs5 logz (fs2 x)+(2*#*x)#'fs4 x}

asinz (2 1.732051 0;0 1 1f) ..
(1.570796 2.153461 3.141593 1.570796 0.9881321 0;1.316958 -1.358838 -0.8813736 -1.316958 1.358838 0.8813736)

arcsin(2)=1.570796 ± 1.316958i ; arcsin(3^1/2+i)=0.9881318+1.358838i,2.153461-1.358838i ; arcsin(i)=0.8813736i,π - 0.8813736i

acosz =-i log(z + i(1 - z²)^1/2)

f(z) = arccos z

acosz:{oo0-asinz x}

acosz (3 0;4 1) ..
(-0.9368125 -1.570796 0.9368125 1.570796;2.305509 0.8813736 -2.305509 -0.8813736)

arccos(3+4i)=± (0.9368125-2.305509i), arccos(i)=± (1.570796-0.8813736i)

atanz = 0.5i log [(i + z)/(i - z)]

f(z) = arctan z

atanz:{c:0.5*fs4 logz fs6 x:ap x;l:last x;c[;&(0w=s:*x)|l in 0w,-0w]:oo0;c[;&-0w=s]:oo1;c[;&(0f=s)&1f< l]*:cp3;if[1=#*c:-1_'c;c:*:'c];:c}

atanz (-5 0 -0w;8 -2 3) .. (-1.514221 -1.570796 -1.570796;0.0898435 -0.5493061 -0)

arctan(-5+8i)=-1.514221+0.0898435i, arctan(-2i)=((-π/2)-0.5493061i) , arctan((-INF)+3i) = -π/2

arsinhz = log (z + (z² + 1)^1/2)

f(z)=arsinh z

arsinhz:{logz (fs1 x)+(2*#*x)#'x}

arsinhz (-100 0 0;5 0 100 ) ..
(5.299591 0 -5.298292 -5.299591 0 5.298292;3.091637 3.141593 1.570796 0.04995591 0 1.570796)

arsinh(-100+5i) = ± 5.299591+3.091637i ,arsinh(0) = 0 or πi, arsinh(100i) = ± 5.298292+πi/2

arcoshz = log (z + (z² - 1)^1/2)

f(z)=arcosh z

arcoshz:{logz (fs3 x)+(2*#*x)#'x}

arcoshz (-100 0 0;5 0 100 ) ..
(5.299541 0 -5.298342 -5.299541 0 5.298342;3.091632 -1.570796 -1.570796 -3.091632 1.570796 1.570796)

arcosh(-100+5i) = ±(5.299591+3.091637i) ,arcosh(0) = 0 ± πi/2, arcosh(100i) = ±(5.298292+πi/2)

artanhz = 0.5 log [(1+z)/(1-z)]

f(z) = artanh z

artanhz:{c:0.5*logz fs7 x:ap x;c[;&(0f<*x)&0f=last x]*:cp2;if[1=#*c:-1_'c;c:*:'c];:c}

artanhz (1 0 1 2f;1 0 -3 0f) .. (0.4023595 0 0.0919312 0.5493061;1.017222 0 -1.276795 -1.570796)

artanh(1+i) = 0.4023595+1.017222i , artanh(0) = 0 , artanh(1-3i) = 0.0919312-1.276795i , artanh(2) = 0.5493061-1.570796i

complex matrix operations

a 1st word

Let be M a complex matrix. We can represent M as Re(M) + i*Im(M) (Form 1) or in the more visible form just as M where each element of M is a pair of real numbers (r,c) (Form 2)
Form 1 to Form 2: tp12:{x,''y}
Form 2 to Form 1: rm:{*:''x} (real part) and cm:{last''x} (complex part)

k)cmult (A;B) , A,B of Form 2

complex matrix multiplication

cmult:{a:(rm x;cm x);b:(rm y;cm y);tp12[((r1 mmu r2)-c1 mmu c2);((r1:a@0) mmu c2:b@1)+(c1:a@1) mmu r2:b@0]}

example:
A:2 2#0f+(2 -1;-3 1;5 3;6 -1) .. ie A[0;0] is 2-i etc.
B:2 2#0f+(3 2;9 -1;6 -3;-1 -1)
A,B are Form 2 matrices. Then C:cmult[A;B] will return the 2x2 matrix 2 2#(-7 16;21 -9;42 -5;41 17) .. ie C[0;0] is -7+16i etc

k)invc M , M of Form 2

complex matrix inversion

invc:{C:(c:#x)#!:0f+((rm x),'cm x),((-1f*cm x),'rm x);:tp12[C[;i];C[;c+i:!c]]}

example:
M:2 2#(2 1;-1 9;7 1;-5 -11) .. ie M is a 2 x 2 complex matrix with M[0;0] = 2+i
Then L:invc M represents M^-1 .. L is then (rounded to 4 decimals) 2 2#(0.1089 -0.0770;0.0996 -0.0078;-0.0037 -0.0780;-0.0067 0.0238) .. ie L[0;0] is 0.1089-0.077i etc

Powerseries Σ _nc_nzⁿ .. n>-1, integer
z, c_j complex numbers

Example 1: Approximate e^0.4+0.7i by a 7-term Powerseries: sum each prdz2 . (flip (1%(*\)1,1_til 7),'0f;firstpow(0.4 0.7;6)) ... 1.140992+0.9610174i

Exact value: e^0.4+0.7i = 1.14101..+0.9610599i

Example 2:

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