diff --git a/libmath/desk/lib/complex.hoon b/libmath/desk/lib/complex.hoon
index 6d76cfb..d9d4b24 100644
--- a/libmath/desk/lib/complex.hoon
+++ b/libmath/desk/lib/complex.hoon
@@ -20,18 +20,17 @@
 ::  additive copies later.  Complex has no total order, so there is
 ::  intentionally no gth/gte/lth/lte -- only equ/neq.
 ::
-::  TODO (PENCILLED IN -- complex transcendentals, follow-on to the %unum
-::  transcendental work).  This lib currently provides arithmetic + abs/conj/
-::  re/im/pak only.  To let Saloon's Tier B (exp/sin/cos/log/sqrt/...) accept
-::  %cplx rays the way it now accepts %unum, each width door needs the complex
-::  elementary functions, then a %cplx branch in saloon +trans-scalar/+fun-scalar
-::  (mirroring the %unum branch).  Closed forms over the real component ops:
-::    cexp(a+bi)  = e^a * (cos b + i sin b)
-::    clog(a+bi)  = log|z| + i*atan2(b, a)            (needs a real atan2)
-::    csqrt(z)    = sqrt((|z|+a)/2) + i*sgn(b)*sqrt((|z|-a)/2)
-::    csin/ccos   via the real sin/cos/sinh/cosh, cpow via cexp(w*clog z)
-::  Depends on the real transcendentals (which #18's Chebyshev rewrite will
-::  harden); accuracy will inherit whatever the real layer provides.
+::  Complex elementary functions (cexp/clog/csqrt/csin/ccos/ctan/cpow) compose
+::  real component transcendentals.  To keep this lib self-contained -- like
+::  /lib/unum, /lib/fixed, /lib/twoc, with no dependency on /lib/math -- each
+::  width door carries its OWN real exp/sin/cos/log/atan as naive Taylor/AGM
+::  series over the stdlib component float door, at a fixed term count (the same
+::  approach as /lib/math and /lib/unum; #18's Chebyshev rewrite would upgrade
+::  those in lockstep).  Closed forms:
+::    cexp(a+bi) = e^a (cos b + i sin b)
+::    clog(a+bi) = log|z| + i atan2(b, a)
+::    csqrt(z)   = sqrt((|z|+a)/2) + i sgn(b) sqrt((|z|-a)/2)
+::    csin/ccos via real sin/cos and cosh/sinh; ctan = csin/ccos; cpow = cexp(w clog z)
 ::
 |%
 +$  rounding-mode  ?(%n %u %d %z)
@@ -202,6 +201,132 @@
   ::      %.y
   ::  Source
   ++  neq   |=([p=@ q=@] =(| (equ p q)))
+  ::
+  ::  Transcendental / elementary functions.  Self-contained real @rs series
+  ::  (naive Taylor/AGM, fixed term counts -- accurate near the expansion point,
+  ::  matching /lib/math and /lib/unum); component arithmetic stays on stdlib +rs.
+  ::
+  ++  rpi   `@rs`.3.1415927                  :: pi at @rs
+  ++  rexp
+    |=  x=@rs  ^-  @rs
+    =/  s  .1  =/  t  .1  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =.  t  (~(mul rs rnd) t (~(div rs rnd) x (~(sun rs rnd) n)))
+    $(n +(n), s (~(add rs rnd) s t))
+  ++  rsin
+    |=  x=@rs  ^-  @rs
+    =/  t  x  =/  s  x  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =/  k  (^mul 2 n)
+    =.  t  (~(sub rs rnd) .0 (~(mul rs rnd) t (~(div rs rnd) (~(mul rs rnd) x x) (~(mul rs rnd) (~(sun rs rnd) k) (~(sun rs rnd) +(k))))))
+    $(n +(n), s (~(add rs rnd) s t))
+  ++  rcos
+    |=  x=@rs  ^-  @rs
+    =/  t  .1  =/  s  .1  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =/  k  (^mul 2 n)
+    =.  t  (~(sub rs rnd) .0 (~(mul rs rnd) t (~(div rs rnd) (~(mul rs rnd) x x) (~(mul rs rnd) (~(sun rs rnd) (dec k)) (~(sun rs rnd) k)))))
+    $(n +(n), s (~(add rs rnd) s t))
+  ++  rlog
+    |=  x=@rs  ^-  @rs
+    ?:  =(`@rs`.0 x)  `@rs`0xff80.0000        :: log 0 = -inf
+    =/  y   (~(div rs rnd) (~(sub rs rnd) x .1) (~(add rs rnd) x .1))
+    =/  y2  (~(mul rs rnd) y y)  =/  s  y  =/  t  y  =/  n=@  1
+    |-  ?:  (gth n 30)  (~(mul rs rnd) .2 s)
+    =.  t  (~(mul rs rnd) t y2)
+    =/  c  (~(div rs rnd) .1 (~(sun rs rnd) +((^mul 2 n))))
+    $(n +(n), s (~(add rs rnd) s (~(mul rs rnd) c t)))
+  ++  ratan                                  :: single-arg arctan (Gauss AGM)
+    |=  x=@rs  ^-  @rs
+    =/  rt  (~(sqt rs rnd) (~(add rs rnd) .1 (~(mul rs rnd) x x)))
+    =/  a  (~(div rs rnd) .1 rt)  =/  b  .1  =/  n=@  0
+    |-  ?:  (gth n 40)  (~(div rs rnd) x (~(mul rs rnd) rt b))
+    =/  ai  (~(mul rs rnd) .0.5 (~(add rs rnd) a b))
+    $(n +(n), a ai, b (~(sqt rs rnd) (~(mul rs rnd) ai b)))
+  ++  ratn                                   :: atan2(y, x)
+    |=  [y=@rs x=@rs]  ^-  @rs
+    ?:  (~(gth rs rnd) x .0)                          (ratan (~(div rs rnd) y x))
+    ?:  &((~(lth rs rnd) x .0) (~(gte rs rnd) y .0))  (~(add rs rnd) (ratan (~(div rs rnd) y x)) rpi)
+    ?:  &((~(lth rs rnd) x .0) (~(lth rs rnd) y .0))  (~(sub rs rnd) (ratan (~(div rs rnd) y x)) rpi)
+    ?:  (~(gth rs rnd) y .0)                          (~(div rs rnd) rpi .2)
+    ?:  (~(lth rs rnd) y .0)                          (~(sub rs rnd) .0 (~(div rs rnd) rpi .2))
+    .0
+  ::    +cexp:  @cs -> @cs
+  ::
+  ::  Complex exponential e^z = e^a (cos b + i sin b) for z = a + bi.
+  ::    Examples
+  ::      > (~(cexp cs:complex %n) 0x4000.0000.3f80.0000)  ::  exp(1+2i)
+  ::      0x401e.30c5.bf90.cb4e
+  ::  Source
+  ++  cexp
+    |=  p=@
+    ^-  @
+    =/  ea  (rexp (re p))
+    (pak (~(mul rs rnd) ea (rcos (im p))) (~(mul rs rnd) ea (rsin (im p))))
+  ::    +clog:  @cs -> @cs
+  ::
+  ::  Principal complex logarithm ln(z) = ln|z| + i*atan2(b, a).
+  ::  Source
+  ++  clog
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  mag  (~(sqt rs rnd) (~(add rs rnd) (~(mul rs rnd) a a) (~(mul rs rnd) b b)))
+    (pak (rlog mag) (ratn b a))
+  ::    +csqrt:  @cs -> @cs
+  ::
+  ::  Principal complex square root: sqrt((|z|+a)/2) + i*sgn(b)*sqrt((|z|-a)/2),
+  ::  with the imaginary sign taken from b (the principal branch).
+  ::  Source
+  ++  csqrt
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  mag  (~(sqt rs rnd) (~(add rs rnd) (~(mul rs rnd) a a) (~(mul rs rnd) b b)))
+    =/  re-  (~(sqt rs rnd) (~(div rs rnd) (~(add rs rnd) mag a) .2))
+    =/  im-  (~(sqt rs rnd) (~(div rs rnd) (~(sub rs rnd) mag a) .2))
+    (pak re- ?:((~(lth rs rnd) b .0) (fneg im-) im-))
+  ::    +csin:  @cs -> @cs
+  ::
+  ::  Complex sine sin(z) = sin(a)cosh(b) + i cos(a)sinh(b), with cosh/sinh built
+  ::  from the real exponential.
+  ::  Source
+  ++  csin
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  eb   (rexp b)
+    =/  enb  (rexp (fneg b))
+    =/  csh  (~(div rs rnd) (~(add rs rnd) eb enb) .2)
+    =/  snh  (~(div rs rnd) (~(sub rs rnd) eb enb) .2)
+    (pak (~(mul rs rnd) (rsin a) csh) (~(mul rs rnd) (rcos a) snh))
+  ::    +ccos:  @cs -> @cs
+  ::
+  ::  Complex cosine cos(z) = cos(a)cosh(b) - i sin(a)sinh(b).
+  ::  Source
+  ++  ccos
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  eb   (rexp b)
+    =/  enb  (rexp (fneg b))
+    =/  csh  (~(div rs rnd) (~(add rs rnd) eb enb) .2)
+    =/  snh  (~(div rs rnd) (~(sub rs rnd) eb enb) .2)
+    (pak (~(mul rs rnd) (rcos a) csh) (fneg (~(mul rs rnd) (rsin a) snh)))
+  ::    +ctan:  @cs -> @cs
+  ::
+  ::  Complex tangent tan(z) = sin(z) / cos(z).
+  ::  Source
+  ++  ctan  |=(p=@ (div (csin p) (ccos p)))
+  ::    +cpow:  [@cs @cs] -> @cs
+  ::
+  ::  Complex power z^w = exp(w * ln z) (principal branch).
+  ::  Source
+  ++  cpow  |=([p=@ q=@] (cexp (mul q (clog p))))
   --
 ::    +cd:  complex-double (@cd), two @rd (64-bit) components.
 ::
@@ -370,6 +495,129 @@
   ::      %.y
   ::  Source
   ++  neq   |=([p=@ q=@] =(| (equ p q)))
+  ::
+  ::  Transcendental / elementary functions.  Self-contained real @rd series
+  ::  (naive Taylor/AGM, fixed term counts -- accurate near the expansion point,
+  ::  matching /lib/math and /lib/unum); component arithmetic stays on stdlib +rs.
+  ::
+  ++  rpi   `@rd`.~3.14159265358979                  :: pi at @rd
+  ++  rexp
+    |=  x=@rd  ^-  @rd
+    =/  s  .~1  =/  t  .~1  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =.  t  (~(mul rd rnd) t (~(div rd rnd) x (~(sun rd rnd) n)))
+    $(n +(n), s (~(add rd rnd) s t))
+  ++  rsin
+    |=  x=@rd  ^-  @rd
+    =/  t  x  =/  s  x  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =/  k  (^mul 2 n)
+    =.  t  (~(sub rd rnd) .~0 (~(mul rd rnd) t (~(div rd rnd) (~(mul rd rnd) x x) (~(mul rd rnd) (~(sun rd rnd) k) (~(sun rd rnd) +(k))))))
+    $(n +(n), s (~(add rd rnd) s t))
+  ++  rcos
+    |=  x=@rd  ^-  @rd
+    =/  t  .~1  =/  s  .~1  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =/  k  (^mul 2 n)
+    =.  t  (~(sub rd rnd) .~0 (~(mul rd rnd) t (~(div rd rnd) (~(mul rd rnd) x x) (~(mul rd rnd) (~(sun rd rnd) (dec k)) (~(sun rd rnd) k)))))
+    $(n +(n), s (~(add rd rnd) s t))
+  ++  rlog
+    |=  x=@rd  ^-  @rd
+    ?:  =(`@rd`.~0 x)  `@rd`0xfff0.0000.0000.0000
+    =/  y   (~(div rd rnd) (~(sub rd rnd) x .~1) (~(add rd rnd) x .~1))
+    =/  y2  (~(mul rd rnd) y y)  =/  s  y  =/  t  y  =/  n=@  1
+    |-  ?:  (gth n 30)  (~(mul rd rnd) .~2 s)
+    =.  t  (~(mul rd rnd) t y2)
+    =/  c  (~(div rd rnd) .~1 (~(sun rd rnd) +((^mul 2 n))))
+    $(n +(n), s (~(add rd rnd) s (~(mul rd rnd) c t)))
+  ++  ratan                                  :: single-arg arctan (Gauss AGM)
+    |=  x=@rd  ^-  @rd
+    =/  rt  (~(sqt rd rnd) (~(add rd rnd) .~1 (~(mul rd rnd) x x)))
+    =/  a  (~(div rd rnd) .~1 rt)  =/  b  .~1  =/  n=@  0
+    |-  ?:  (gth n 40)  (~(div rd rnd) x (~(mul rd rnd) rt b))
+    =/  ai  (~(mul rd rnd) .~0.5 (~(add rd rnd) a b))
+    $(n +(n), a ai, b (~(sqt rd rnd) (~(mul rd rnd) ai b)))
+  ++  ratn                                   :: atan2(y, x)
+    |=  [y=@rd x=@rd]  ^-  @rd
+    ?:  (~(gth rd rnd) x .~0)                          (ratan (~(div rd rnd) y x))
+    ?:  &((~(lth rd rnd) x .~0) (~(gte rd rnd) y .~0))  (~(add rd rnd) (ratan (~(div rd rnd) y x)) rpi)
+    ?:  &((~(lth rd rnd) x .~0) (~(lth rd rnd) y .~0))  (~(sub rd rnd) (ratan (~(div rd rnd) y x)) rpi)
+    ?:  (~(gth rd rnd) y .~0)                          (~(div rd rnd) rpi .~2)
+    ?:  (~(lth rd rnd) y .~0)                          (~(sub rd rnd) .~0 (~(div rd rnd) rpi .~2))
+    .~0
+  ::    +cexp:  @cd -> @cd
+  ::
+  ::  Complex exponential e^z = e^a (cos b + i sin b) for z = a + bi.
+  ::  Source
+  ++  cexp
+    |=  p=@
+    ^-  @
+    =/  ea  (rexp (re p))
+    (pak (~(mul rd rnd) ea (rcos (im p))) (~(mul rd rnd) ea (rsin (im p))))
+  ::    +clog:  @cd -> @cd
+  ::
+  ::  Principal complex logarithm ln(z) = ln|z| + i*atan2(b, a).
+  ::  Source
+  ++  clog
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  mag  (~(sqt rd rnd) (~(add rd rnd) (~(mul rd rnd) a a) (~(mul rd rnd) b b)))
+    (pak (rlog mag) (ratn b a))
+  ::    +csqrt:  @cd -> @cd
+  ::
+  ::  Principal complex square root: sqrt((|z|+a)/2) + i*sgn(b)*sqrt((|z|-a)/2),
+  ::  with the imaginary sign taken from b (the principal branch).
+  ::  Source
+  ++  csqrt
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  mag  (~(sqt rd rnd) (~(add rd rnd) (~(mul rd rnd) a a) (~(mul rd rnd) b b)))
+    =/  re-  (~(sqt rd rnd) (~(div rd rnd) (~(add rd rnd) mag a) .~2))
+    =/  im-  (~(sqt rd rnd) (~(div rd rnd) (~(sub rd rnd) mag a) .~2))
+    (pak re- ?:((~(lth rd rnd) b .~0) (fneg im-) im-))
+  ::    +csin:  @cd -> @cd
+  ::
+  ::  Complex sine sin(z) = sin(a)cosh(b) + i cos(a)sinh(b), with cosh/sinh built
+  ::  from the real exponential.
+  ::  Source
+  ++  csin
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  eb   (rexp b)
+    =/  enb  (rexp (fneg b))
+    =/  csh  (~(div rd rnd) (~(add rd rnd) eb enb) .~2)
+    =/  snh  (~(div rd rnd) (~(sub rd rnd) eb enb) .~2)
+    (pak (~(mul rd rnd) (rsin a) csh) (~(mul rd rnd) (rcos a) snh))
+  ::    +ccos:  @cd -> @cd
+  ::
+  ::  Complex cosine cos(z) = cos(a)cosh(b) - i sin(a)sinh(b).
+  ::  Source
+  ++  ccos
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  eb   (rexp b)
+    =/  enb  (rexp (fneg b))
+    =/  csh  (~(div rd rnd) (~(add rd rnd) eb enb) .~2)
+    =/  snh  (~(div rd rnd) (~(sub rd rnd) eb enb) .~2)
+    (pak (~(mul rd rnd) (rcos a) csh) (fneg (~(mul rd rnd) (rsin a) snh)))
+  ::    +ctan:  @cd -> @cd
+  ::
+  ::  Complex tangent tan(z) = sin(z) / cos(z).
+  ::  Source
+  ++  ctan  |=(p=@ (div (csin p) (ccos p)))
+  ::    +cpow:  [@cd @cd] -> @cd
+  ::
+  ::  Complex power z^w = exp(w * ln z) (principal branch).
+  ::  Source
+  ++  cpow  |=([p=@ q=@] (cexp (mul q (clog p))))
   --
 ::    +ch:  complex-half (@ch), two @rh (16-bit) components.
 ::
@@ -536,6 +784,129 @@
   ::      %.y
   ::  Source
   ++  neq   |=([p=@ q=@] =(| (equ p q)))
+  ::
+  ::  Transcendental / elementary functions.  Self-contained real @rh series
+  ::  (naive Taylor/AGM, fixed term counts -- accurate near the expansion point,
+  ::  matching /lib/math and /lib/unum); component arithmetic stays on stdlib +rs.
+  ::
+  ++  rpi   `@rh`.~~3.14159                  :: pi at @rh
+  ++  rexp
+    |=  x=@rh  ^-  @rh
+    =/  s  .~~1  =/  t  .~~1  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =.  t  (~(mul rh rnd) t (~(div rh rnd) x (~(sun rh rnd) n)))
+    $(n +(n), s (~(add rh rnd) s t))
+  ++  rsin
+    |=  x=@rh  ^-  @rh
+    =/  t  x  =/  s  x  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =/  k  (^mul 2 n)
+    =.  t  (~(sub rh rnd) .~~0 (~(mul rh rnd) t (~(div rh rnd) (~(mul rh rnd) x x) (~(mul rh rnd) (~(sun rh rnd) k) (~(sun rh rnd) +(k))))))
+    $(n +(n), s (~(add rh rnd) s t))
+  ++  rcos
+    |=  x=@rh  ^-  @rh
+    =/  t  .~~1  =/  s  .~~1  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =/  k  (^mul 2 n)
+    =.  t  (~(sub rh rnd) .~~0 (~(mul rh rnd) t (~(div rh rnd) (~(mul rh rnd) x x) (~(mul rh rnd) (~(sun rh rnd) (dec k)) (~(sun rh rnd) k)))))
+    $(n +(n), s (~(add rh rnd) s t))
+  ++  rlog
+    |=  x=@rh  ^-  @rh
+    ?:  =(`@rh`.~~0 x)  `@rh`0xfc00
+    =/  y   (~(div rh rnd) (~(sub rh rnd) x .~~1) (~(add rh rnd) x .~~1))
+    =/  y2  (~(mul rh rnd) y y)  =/  s  y  =/  t  y  =/  n=@  1
+    |-  ?:  (gth n 30)  (~(mul rh rnd) .~~2 s)
+    =.  t  (~(mul rh rnd) t y2)
+    =/  c  (~(div rh rnd) .~~1 (~(sun rh rnd) +((^mul 2 n))))
+    $(n +(n), s (~(add rh rnd) s (~(mul rh rnd) c t)))
+  ++  ratan                                  :: single-arg arctan (Gauss AGM)
+    |=  x=@rh  ^-  @rh
+    =/  rt  (~(sqt rh rnd) (~(add rh rnd) .~~1 (~(mul rh rnd) x x)))
+    =/  a  (~(div rh rnd) .~~1 rt)  =/  b  .~~1  =/  n=@  0
+    |-  ?:  (gth n 40)  (~(div rh rnd) x (~(mul rh rnd) rt b))
+    =/  ai  (~(mul rh rnd) .~~0.5 (~(add rh rnd) a b))
+    $(n +(n), a ai, b (~(sqt rh rnd) (~(mul rh rnd) ai b)))
+  ++  ratn                                   :: atan2(y, x)
+    |=  [y=@rh x=@rh]  ^-  @rh
+    ?:  (~(gth rh rnd) x .~~0)                          (ratan (~(div rh rnd) y x))
+    ?:  &((~(lth rh rnd) x .~~0) (~(gte rh rnd) y .~~0))  (~(add rh rnd) (ratan (~(div rh rnd) y x)) rpi)
+    ?:  &((~(lth rh rnd) x .~~0) (~(lth rh rnd) y .~~0))  (~(sub rh rnd) (ratan (~(div rh rnd) y x)) rpi)
+    ?:  (~(gth rh rnd) y .~~0)                          (~(div rh rnd) rpi .~~2)
+    ?:  (~(lth rh rnd) y .~~0)                          (~(sub rh rnd) .~~0 (~(div rh rnd) rpi .~~2))
+    .~~0
+  ::    +cexp:  @ch -> @ch
+  ::
+  ::  Complex exponential e^z = e^a (cos b + i sin b) for z = a + bi.
+  ::  Source
+  ++  cexp
+    |=  p=@
+    ^-  @
+    =/  ea  (rexp (re p))
+    (pak (~(mul rh rnd) ea (rcos (im p))) (~(mul rh rnd) ea (rsin (im p))))
+  ::    +clog:  @ch -> @ch
+  ::
+  ::  Principal complex logarithm ln(z) = ln|z| + i*atan2(b, a).
+  ::  Source
+  ++  clog
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  mag  (~(sqt rh rnd) (~(add rh rnd) (~(mul rh rnd) a a) (~(mul rh rnd) b b)))
+    (pak (rlog mag) (ratn b a))
+  ::    +csqrt:  @ch -> @ch
+  ::
+  ::  Principal complex square root: sqrt((|z|+a)/2) + i*sgn(b)*sqrt((|z|-a)/2),
+  ::  with the imaginary sign taken from b (the principal branch).
+  ::  Source
+  ++  csqrt
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  mag  (~(sqt rh rnd) (~(add rh rnd) (~(mul rh rnd) a a) (~(mul rh rnd) b b)))
+    =/  re-  (~(sqt rh rnd) (~(div rh rnd) (~(add rh rnd) mag a) .~~2))
+    =/  im-  (~(sqt rh rnd) (~(div rh rnd) (~(sub rh rnd) mag a) .~~2))
+    (pak re- ?:((~(lth rh rnd) b .~~0) (fneg im-) im-))
+  ::    +csin:  @ch -> @ch
+  ::
+  ::  Complex sine sin(z) = sin(a)cosh(b) + i cos(a)sinh(b), with cosh/sinh built
+  ::  from the real exponential.
+  ::  Source
+  ++  csin
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  eb   (rexp b)
+    =/  enb  (rexp (fneg b))
+    =/  csh  (~(div rh rnd) (~(add rh rnd) eb enb) .~~2)
+    =/  snh  (~(div rh rnd) (~(sub rh rnd) eb enb) .~~2)
+    (pak (~(mul rh rnd) (rsin a) csh) (~(mul rh rnd) (rcos a) snh))
+  ::    +ccos:  @ch -> @ch
+  ::
+  ::  Complex cosine cos(z) = cos(a)cosh(b) - i sin(a)sinh(b).
+  ::  Source
+  ++  ccos
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  eb   (rexp b)
+    =/  enb  (rexp (fneg b))
+    =/  csh  (~(div rh rnd) (~(add rh rnd) eb enb) .~~2)
+    =/  snh  (~(div rh rnd) (~(sub rh rnd) eb enb) .~~2)
+    (pak (~(mul rh rnd) (rcos a) csh) (fneg (~(mul rh rnd) (rsin a) snh)))
+  ::    +ctan:  @ch -> @ch
+  ::
+  ::  Complex tangent tan(z) = sin(z) / cos(z).
+  ::  Source
+  ++  ctan  |=(p=@ (div (csin p) (ccos p)))
+  ::    +cpow:  [@ch @ch] -> @ch
+  ::
+  ::  Complex power z^w = exp(w * ln z) (principal branch).
+  ::  Source
+  ++  cpow  |=([p=@ q=@] (cexp (mul q (clog p))))
   --
 ::    +cq:  complex-quad (@cq), two @rq (128-bit) components.
 ::
@@ -703,5 +1074,128 @@
   ::      %.y
   ::  Source
   ++  neq   |=([p=@ q=@] =(| (equ p q)))
+  ::
+  ::  Transcendental / elementary functions.  Self-contained real @rq series
+  ::  (naive Taylor/AGM, fixed term counts -- accurate near the expansion point,
+  ::  matching /lib/math and /lib/unum); component arithmetic stays on stdlib +rs.
+  ::
+  ++  rpi   `@rq`.~~~3.14159265358979323846                  :: pi at @rq
+  ++  rexp
+    |=  x=@rq  ^-  @rq
+    =/  s  .~~~1  =/  t  .~~~1  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =.  t  (~(mul rq rnd) t (~(div rq rnd) x (~(sun rq rnd) n)))
+    $(n +(n), s (~(add rq rnd) s t))
+  ++  rsin
+    |=  x=@rq  ^-  @rq
+    =/  t  x  =/  s  x  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =/  k  (^mul 2 n)
+    =.  t  (~(sub rq rnd) .~~~0 (~(mul rq rnd) t (~(div rq rnd) (~(mul rq rnd) x x) (~(mul rq rnd) (~(sun rq rnd) k) (~(sun rq rnd) +(k))))))
+    $(n +(n), s (~(add rq rnd) s t))
+  ++  rcos
+    |=  x=@rq  ^-  @rq
+    =/  t  .~~~1  =/  s  .~~~1  =/  n=@  1
+    |-  ?:  (gth n 20)  s
+    =/  k  (^mul 2 n)
+    =.  t  (~(sub rq rnd) .~~~0 (~(mul rq rnd) t (~(div rq rnd) (~(mul rq rnd) x x) (~(mul rq rnd) (~(sun rq rnd) (dec k)) (~(sun rq rnd) k)))))
+    $(n +(n), s (~(add rq rnd) s t))
+  ++  rlog
+    |=  x=@rq  ^-  @rq
+    ?:  =(`@rq`.~~~0 x)  `@rq`0xffff.0000.0000.0000.0000.0000.0000.0000
+    =/  y   (~(div rq rnd) (~(sub rq rnd) x .~~~1) (~(add rq rnd) x .~~~1))
+    =/  y2  (~(mul rq rnd) y y)  =/  s  y  =/  t  y  =/  n=@  1
+    |-  ?:  (gth n 30)  (~(mul rq rnd) .~~~2 s)
+    =.  t  (~(mul rq rnd) t y2)
+    =/  c  (~(div rq rnd) .~~~1 (~(sun rq rnd) +((^mul 2 n))))
+    $(n +(n), s (~(add rq rnd) s (~(mul rq rnd) c t)))
+  ++  ratan                                  :: single-arg arctan (Gauss AGM)
+    |=  x=@rq  ^-  @rq
+    =/  rt  (~(sqt rq rnd) (~(add rq rnd) .~~~1 (~(mul rq rnd) x x)))
+    =/  a  (~(div rq rnd) .~~~1 rt)  =/  b  .~~~1  =/  n=@  0
+    |-  ?:  (gth n 40)  (~(div rq rnd) x (~(mul rq rnd) rt b))
+    =/  ai  (~(mul rq rnd) .~~~0.5 (~(add rq rnd) a b))
+    $(n +(n), a ai, b (~(sqt rq rnd) (~(mul rq rnd) ai b)))
+  ++  ratn                                   :: atan2(y, x)
+    |=  [y=@rq x=@rq]  ^-  @rq
+    ?:  (~(gth rq rnd) x .~~~0)                          (ratan (~(div rq rnd) y x))
+    ?:  &((~(lth rq rnd) x .~~~0) (~(gte rq rnd) y .~~~0))  (~(add rq rnd) (ratan (~(div rq rnd) y x)) rpi)
+    ?:  &((~(lth rq rnd) x .~~~0) (~(lth rq rnd) y .~~~0))  (~(sub rq rnd) (ratan (~(div rq rnd) y x)) rpi)
+    ?:  (~(gth rq rnd) y .~~~0)                          (~(div rq rnd) rpi .~~~2)
+    ?:  (~(lth rq rnd) y .~~~0)                          (~(sub rq rnd) .~~~0 (~(div rq rnd) rpi .~~~2))
+    .~~~0
+  ::    +cexp:  @cq -> @cq
+  ::
+  ::  Complex exponential e^z = e^a (cos b + i sin b) for z = a + bi.
+  ::  Source
+  ++  cexp
+    |=  p=@
+    ^-  @
+    =/  ea  (rexp (re p))
+    (pak (~(mul rq rnd) ea (rcos (im p))) (~(mul rq rnd) ea (rsin (im p))))
+  ::    +clog:  @cq -> @cq
+  ::
+  ::  Principal complex logarithm ln(z) = ln|z| + i*atan2(b, a).
+  ::  Source
+  ++  clog
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  mag  (~(sqt rq rnd) (~(add rq rnd) (~(mul rq rnd) a a) (~(mul rq rnd) b b)))
+    (pak (rlog mag) (ratn b a))
+  ::    +csqrt:  @cq -> @cq
+  ::
+  ::  Principal complex square root: sqrt((|z|+a)/2) + i*sgn(b)*sqrt((|z|-a)/2),
+  ::  with the imaginary sign taken from b (the principal branch).
+  ::  Source
+  ++  csqrt
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  mag  (~(sqt rq rnd) (~(add rq rnd) (~(mul rq rnd) a a) (~(mul rq rnd) b b)))
+    =/  re-  (~(sqt rq rnd) (~(div rq rnd) (~(add rq rnd) mag a) .~~~2))
+    =/  im-  (~(sqt rq rnd) (~(div rq rnd) (~(sub rq rnd) mag a) .~~~2))
+    (pak re- ?:((~(lth rq rnd) b .~~~0) (fneg im-) im-))
+  ::    +csin:  @cq -> @cq
+  ::
+  ::  Complex sine sin(z) = sin(a)cosh(b) + i cos(a)sinh(b), with cosh/sinh built
+  ::  from the real exponential.
+  ::  Source
+  ++  csin
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  eb   (rexp b)
+    =/  enb  (rexp (fneg b))
+    =/  csh  (~(div rq rnd) (~(add rq rnd) eb enb) .~~~2)
+    =/  snh  (~(div rq rnd) (~(sub rq rnd) eb enb) .~~~2)
+    (pak (~(mul rq rnd) (rsin a) csh) (~(mul rq rnd) (rcos a) snh))
+  ::    +ccos:  @cq -> @cq
+  ::
+  ::  Complex cosine cos(z) = cos(a)cosh(b) - i sin(a)sinh(b).
+  ::  Source
+  ++  ccos
+    |=  p=@
+    ^-  @
+    =/  a  (re p)
+    =/  b  (im p)
+    =/  eb   (rexp b)
+    =/  enb  (rexp (fneg b))
+    =/  csh  (~(div rq rnd) (~(add rq rnd) eb enb) .~~~2)
+    =/  snh  (~(div rq rnd) (~(sub rq rnd) eb enb) .~~~2)
+    (pak (~(mul rq rnd) (rcos a) csh) (fneg (~(mul rq rnd) (rsin a) snh)))
+  ::    +ctan:  @cq -> @cq
+  ::
+  ::  Complex tangent tan(z) = sin(z) / cos(z).
+  ::  Source
+  ++  ctan  |=(p=@ (div (csin p) (ccos p)))
+  ::    +cpow:  [@cq @cq] -> @cq
+  ::
+  ::  Complex power z^w = exp(w * ln z) (principal branch).
+  ::  Source
+  ++  cpow  |=([p=@ q=@] (cexp (mul q (clog p))))
   --
 --
diff --git a/libmath/desk/tests/lib/complex-edge.hoon b/libmath/desk/tests/lib/complex-edge.hoon
new file mode 100644
index 0000000..b8b9e94
--- /dev/null
+++ b/libmath/desk/tests/lib/complex-edge.hoon
@@ -0,0 +1,99 @@
+::  /tests/lib/complex-edge -- %cplx transcendental edge cases (@cs).
+::
+::  Exercises the origin, the real and imaginary axes, the csqrt/clog branch cut
+::  (-1, +-i), the csqrt sign branch, the clog(0) singularity, and the naive-
+::  series breakdown at large arguments.  All values validated against numpy
+::  complex64.
+::
+/+  *test, complex
+|%
+++  s    cs:complex
+++  z00  `@`0x0                    ::  0+0i
+++  z10  `@`0x3f80.0000            ::  1+0i
+++  zt0  `@`0x4120.0000            ::  10+0i
+++  z0i  `@`0x3f80.0000.0000.0000  ::  0+1i
+++  z0n  `@`0xbf80.0000.0000.0000  ::  0-1i
+++  zn1  `@`0xbf80.0000            ::  -1+0i
+++  z11  `@`0x3f80.0000.3f80.0000  ::  1+1i
+++  z34  `@`0x4080.0000.4040.0000  ::  3+4i
+++  z0t  `@`0x4120.0000.0000.0000  ::  0+10i
+::  cexp: origin -> 1, real axis (e, 1/e), imaginary axis (cos1 + i sin1), off-axis.
+++  test-cexp-origin  (expect-eq !>(`@`0x3f80.0000) !>((~(cexp s %n) z00)))
+++  test-cexp-1       (expect-eq !>(`@`0x402d.f855) !>((~(cexp s %n) z10)))
+++  test-cexp-neg1    (expect-eq !>(`@`0x3ebc.5ab0) !>((~(cexp s %n) zn1)))
+++  test-cexp-i       (expect-eq !>(`@`0x3f57.6aa4.3f0a.5140) !>((~(cexp s %n) z0i)))
+++  test-cexp-1p1i    (expect-eq !>(`@`0x4012.6408.3fbb.fe2a) !>((~(cexp s %n) z11)))
+++  test-cexp-3p4i    (expect-eq !>(`@`0xc173.366f.c152.0f81) !>((~(cexp s %n) z34)))
+::  clog: the singularity (log 0 = -inf) and the branch cut at -1 (iπ) and +-i.
+++  test-clog-zero    (expect-eq !>(`@`0xff80.0000) !>((~(clog s %n) z00)))
+++  test-clog-neg1    (expect-eq !>(`@`0x4049.0fdb.0000.0000) !>((~(clog s %n) zn1)))
+++  test-clog-i       (expect-eq !>(`@`0x3fc9.0fdb.0000.0000) !>((~(clog s %n) z0i)))
+++  test-clog-neg-i   (expect-eq !>(`@`0xbfc9.0fdb.0000.0000) !>((~(clog s %n) z0n)))
+::  csqrt: branch cut sqrt(-1) = i, and the sign branch sqrt(-i) = 0.707 - 0.707i.
+++  test-csqrt-neg1   (expect-eq !>(`@`0x3f80.0000.0000.0000) !>((~(csqrt s %n) zn1)))
+++  test-csqrt-neg-i  (expect-eq !>(`@`0xbf35.04f3.3f35.04f3) !>((~(csqrt s %n) z0n)))
+::  csin/ccos on the imaginary axis -> real sinh/cosh.
+++  test-csin-i       (expect-eq !>(`@`0x3f96.6cff.0000.0000) !>((~(csin s %n) z0i)))
+++  test-ccos-i       (expect-eq !>(`@`0x3fc5.83ab) !>((~(ccos s %n) z0i)))
+::  KNOWN LIMITATION: the naive Taylor/AGM series diverges from the true value
+::  far from the origin -- exp(10) is ~21991 here vs the true ~22026 (~0.16%
+::  low), csin(0+10i) ~ i*10989 vs true ~ i*11013.  Locked as regression (NOT as
+::  correctness); #18's Chebyshev rewrite would tighten these.
+++  test-cexp-large   (expect-eq !>(`@`0x46ab.cef6) !>((~(cexp s %n) zt0)))
+++  test-csin-large   (expect-eq !>(`@`0x462b.b42b.0000.0000) !>((~(csin s %n) z0t)))
+::
+::  Same edges across the other widths: @cd (double), @ch (half), @cq (quad).
+::  Origin, the csqrt/clog branch cut at -1, the clog(0)=-inf singularity, the
+::  csqrt sign branch at -i, and the imaginary axis (csin -> sinh).
+++  d   cd:complex
+++  h   ch:complex
+++  q   cq:complex
+++  d00  `@`0x0
+++  dn1  `@`0xbff0.0000.0000.0000
+++  d0i  `@`0x3ff0.0000.0000.0000.0000.0000.0000.0000
+++  d0n  `@`0xbff0.0000.0000.0000.0000.0000.0000.0000
+++  h00  `@`0x0
+++  hn1  `@`0xbc00
+++  h0i  `@`0x3c00.0000
+++  h0n  `@`0xbc00.0000
+++  q00  `@`0x0
+++  qn1  `@`0xbfff.0000.0000.0000.0000.0000.0000.0000
+++  test-cd-cexp-origin  (expect-eq !>(`@`0x3ff0.0000.0000.0000) !>((~(cexp d %n) d00)))
+++  test-cd-csqrt-neg1   (expect-eq !>(`@`0x3ff0.0000.0000.0000.0000.0000.0000.0000) !>((~(csqrt d %n) dn1)))
+++  test-cd-clog-neg1    (expect-eq !>(`@`0x4009.21fb.5444.2d11.0000.0000.0000.0000) !>((~(clog d %n) dn1)))
+++  test-cd-clog-zero    (expect-eq !>(`@`0xfff0.0000.0000.0000) !>((~(clog d %n) d00)))
+++  test-cd-csqrt-neg-i  (expect-eq !>(`@`0xbfe6.a09e.667f.3bcd.3fe6.a09e.667f.3bcd) !>((~(csqrt d %n) d0n)))
+++  test-cd-csin-i       (expect-eq !>(`@`0x3ff2.cd9f.c44e.b983.0000.0000.0000.0000) !>((~(csin d %n) d0i)))
+++  test-ch-cexp-origin  (expect-eq !>(`@`0x3c00) !>((~(cexp h %n) h00)))
+++  test-ch-csqrt-neg1   (expect-eq !>(`@`0x3c00.0000) !>((~(csqrt h %n) hn1)))
+++  test-ch-clog-zero    (expect-eq !>(`@`0xfc00) !>((~(clog h %n) h00)))
+++  test-ch-csqrt-neg-i  (expect-eq !>(`@`0xb9a8.39a8) !>((~(csqrt h %n) h0n)))
+++  test-ch-csin-i       (expect-eq !>(`@`0x3cb2.0000) !>((~(csin h %n) h0i)))
+++  test-cq-cexp-origin  (expect-eq !>(`@`0x3fff.0000.0000.0000.0000.0000.0000.0000) !>((~(cexp q %n) q00)))
+++  test-cq-csqrt-neg1
+  (expect-eq !>(`@`0x3fff.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000) !>((~(csqrt q %n) qn1)))
+++  test-cq-clog-zero    (expect-eq !>(`@`0xffff.0000.0000.0000.0000.0000.0000.0000) !>((~(clog q %n) q00)))
+::
+::  cpow edge cases (@cs): z^0 = 1, integer powers, the branch sqrt via ^0.5,
+::  and the 0^w corner.
+++  cs2   `@`0x4000.0000               ::  2+0i
+++  csi   `@`0x3f80.0000.0000.0000     ::  0+1i
+++  csn1  `@`0xbf80.0000               ::  -1+0i
+++  cwh   `@`0x3f00.0000               ::  0.5+0i
+++  test-cpow-pow0      (expect-eq !>(`@`0x3f80.0000) !>((~(cpow s %n) cs2 z00)))         ::  2^0 = 1
+++  test-cpow-pow2      (expect-eq !>(`@`0x4080.0000) !>((~(cpow s %n) cs2 cs2)))         ::  2^2 = 4
+++  test-cpow-i2        (expect-eq !>(`@`0xb182.92c0.bf80.0000) !>((~(cpow s %n) csi cs2)))   ::  i^2 = -1 (+~0i)
+++  test-cpow-neg1-half  (expect-eq !>(`@`0x3f7f.ffff.b386.2919) !>((~(cpow s %n) csn1 cwh)))  ::  (-1)^0.5 = i
+::  KNOWN LIMITATION: 0^2 should be 0, but cpow = exp(w*clog z) routes through
+::  clog(0)=-inf and exp(-inf) (which the naive series cannot represent), so it
+::  yields NaN.  Locked as a documented corner, not as correctness.
+++  test-cpow-zero-pow2  (expect-eq !>(`@`0x7fc0.0000.7fc0.0000) !>((~(cpow s %n) z00 cs2)))
+::
+::  Rounding modes propagate through the complex ops: cexp(1+2i) gives four
+::  distinct bit patterns under %n / %u / %d / %z.
+++  z12  `@`0x4000.0000.3f80.0000      ::  1+2i
+++  test-cexp-rnd-n  (expect-eq !>(`@`0x401e.30c5.bf90.cb4e) !>((~(cexp s %n) z12)))
+++  test-cexp-rnd-u  (expect-eq !>(`@`0x401e.30d9.bf90.cb53) !>((~(cexp s %u) z12)))
+++  test-cexp-rnd-d  (expect-eq !>(`@`0x401e.30bd.bf90.cb53) !>((~(cexp s %d) z12)))
+++  test-cexp-rnd-z  (expect-eq !>(`@`0x401e.30bd.bf90.cb46) !>((~(cexp s %z) z12)))
+--
diff --git a/libmath/desk/tests/lib/complex-fns.hoon b/libmath/desk/tests/lib/complex-fns.hoon
new file mode 100644
index 0000000..024b395
--- /dev/null
+++ b/libmath/desk/tests/lib/complex-fns.hoon
@@ -0,0 +1,35 @@
+::  /tests/lib/complex-fns -- complex elementary functions.
+::
+::  Expected values are the lib's own self-contained series outputs, validated
+::  against numpy (complex64 / complex128) to display precision, e.g.
+::  cexp(1+2i) = -1.1312 + 2.4717i, clog = 0.8047 + 1.1071i, csqrt = 1.2720 +
+::  0.7862i, csin = 3.1658 + 1.9596i, ccos = 2.0327 - 3.0519i, ctan = 0.0338 +
+::  1.0148i, (1+2i)^2 = -3 + 4i.  Inputs: z = 1+2i at each width.
+::
+/+  *test, complex
+|%
+++  s   cs:complex
+++  d   cd:complex
+++  h   ch:complex
+++  q   cq:complex
+++  zs  `@`0x4000.0000.3f80.0000
+++  zd  `@`0x4000.0000.0000.0000.3ff0.0000.0000.0000
+++  zh  `@`0x4000.3c00
+++  zq  `@`0x4000.0000.0000.0000.0000.0000.0000.0000.3fff.0000.0000.0000.0000.0000.0000.0000
+::  @cs (single): the full set; cpow uses w = 2+0i, so (1+2i)^2 = -3+4i.
+++  test-cs-cexp   (expect-eq !>(`@`0x401e.30c5.bf90.cb4e) !>((~(cexp s %n) zs)))
+++  test-cs-clog   (expect-eq !>(`@`0x3f8d.b70a.3f4e.0210) !>((~(clog s %n) zs)))
+++  test-cs-csqrt  (expect-eq !>(`@`0x3f49.4138.3fa2.d18a) !>((~(csqrt s %n) zs)))
+++  test-cs-csin   (expect-eq !>(`@`0x3ffa.d435.404a.9c1e) !>((~(csin s %n) zs)))
+++  test-cs-ccos   (expect-eq !>(`@`0xc043.524b.4002.1823) !>((~(ccos s %n) zs)))
+++  test-cs-ctan   (expect-eq !>(`@`0x3f81.e4c1.3d0a.7f5c) !>((~(ctan s %n) zs)))
+++  test-cs-cpow   (expect-eq !>(`@`0x4080.0004.c03f.fff5) !>((~(cpow s %n) zs `@`0x4000.0000)))
+::  @cd (double): exp/log/sqrt -- matches numpy complex128 to ~15 digits.
+++  test-cd-cexp   (expect-eq !>(`@`0x4003.c618.a227.4afe.bff2.1969.c495.3cd3) !>((~(cexp d %n) zd)))
+++  test-cd-clog   (expect-eq !>(`@`0x3ff1.b6e1.92eb.be45.3fe9.c041.f7ed.8d33) !>((~(clog d %n) zd)))
+++  test-cd-csqrt  (expect-eq !>(`@`0x3fe9.2826.ef25.8d1b.3ff4.5a31.46a8.8456) !>((~(csqrt d %n) zd)))
+::  @ch (half) + @cq (quad): cexp spot check across the remaining widths.
+++  test-ch-cexp   (expect-eq !>(`@`0x40f1.bc86) !>((~(cexp h %n) zh)))
+++  test-cq-cexp
+  (expect-eq !>(`@`0x4000.3c61.8a22.74af.d5ad.4589.2de9.748d.bfff.2196.9c49.53cd.175c.75fb.0c1e.697d) !>((~(cexp q %n) zq)))
+--
diff --git a/libmath/desk/tests/lib/unum-edge.hoon b/libmath/desk/tests/lib/unum-edge.hoon
new file mode 100644
index 0000000..92729e7
--- /dev/null
+++ b/libmath/desk/tests/lib/unum-edge.hoon
@@ -0,0 +1,59 @@
+::  /tests/lib/unum-edge -- posit (%unum) transcendental edge cases.
+::
+::  Exact identities, sign, NaR propagation, domain -> NaR, and the naive-series
+::  breakdown / saturation at large arguments.  Mostly at posit32 (rps), with a
+::  posit8 (rpb) cross-width spot.  Posits round-to-nearest-even, so there is no
+::  rounding-mode variation to sweep.
+::
+/+  *test, unum
+|%
+++  u    rps:unum
+++  b    rpb:unum
+++  nar  `@`0x8000.0000
+++  s1   (sun:rps:unum 1)
+++  s4   (sun:rps:unum 4)
+++  s5   (sun:rps:unum 5)
+++  sa   (sun:rps:unum 10)
+++  sb   (sun:rps:unum 50)
+++  sc   (sun:rps:unum 100)
+++  n1   (neg:rps:unum (sun:rps:unum 1))
+::  exact identities (correctly rounded; series accurate near 0)
+++  test-exp0    (expect-eq !>(`@`0x4000.0000) !>((exp:u 0x0)))         ::  exp 0 = 1
+++  test-cos0    (expect-eq !>(`@`0x4000.0000) !>((cos:u 0x0)))         ::  cos 0 = 1
+++  test-sin0    (expect-eq !>(`@`0x0) !>((sin:u 0x0)))                 ::  sin 0 = 0
+++  test-tan0    (expect-eq !>(`@`0x0) !>((tan:u 0x0)))
+++  test-log1    (expect-eq !>(`@`0x0) !>((log:u s1)))                  ::  log 1 = 0
+++  test-sqt0    (expect-eq !>(`@`0x0) !>((sqt:u 0x0)))
+++  test-sqt1    (expect-eq !>(`@`0x4000.0000) !>((sqt:u s1)))
+++  test-sqt4    (expect-eq !>(`@`0x4800.0000) !>((sqt:u s4)))          ::  sqrt 4 = 2
+++  test-fact0   (expect-eq !>(`@`0x4000.0000) !>((factorial:u 0x0)))   ::  0! = 1
+++  test-fact5   (expect-eq !>(`@`0x6b80.0000) !>((factorial:u s5)))    ::  5! = 120
+++  test-atan0   (expect-eq !>(`@`0x0) !>((atan:u 0x0)))
+++  test-asin0   (expect-eq !>(`@`0x0) !>((asin:u 0x0)))
+++  test-acos1   (expect-eq !>(`@`0x0) !>((acos:u s1)))                 ::  acos 1 = 0
+::  sign
+++  test-exp-n1   (expect-eq !>(`@`0x33c5.ab1c) !>((exp:u n1)))         ::  1/e
+++  test-atan-n1  (expect-eq !>(`@`0xc36f.0255) !>((atan:u n1)))        ::  -pi/4 (atan is odd)
+::  NaR propagation
+++  test-exp-nar   (expect-eq !>(`@`0x8000.0000) !>((exp:u nar)))
+++  test-sin-nar   (expect-eq !>(`@`0x8000.0000) !>((sin:u nar)))
+++  test-sqt-nar   (expect-eq !>(`@`0x8000.0000) !>((sqt:u nar)))
+++  test-fact-nar  (expect-eq !>(`@`0x8000.0000) !>((factorial:u nar)))
+::  domain -> NaR
+++  test-sqt-neg  (expect-eq !>(`@`0x8000.0000) !>((sqt:u n1)))         ::  sqrt(-1) = NaR
+::  posit8 cross-width spot: identity + NaR propagation (-1 @rpb = 0xc0)
+++  test-rpb-exp0     (expect-eq !>(`@`0x40) !>((exp:b 0x0)))           ::  exp 0 = 1
+++  test-rpb-exp-nar  (expect-eq !>(`@`0x80) !>((exp:b 0x80)))
+++  test-rpb-sqt-neg  (expect-eq !>(`@`0x80) !>((sqt:b 0xc0)))          ::  sqrt(-1) = NaR
+::  KNOWN LIMITATION: the naive Taylor series diverges far from the origin.
+::  exp(10) ~ 21991 vs true 22026; exp(50)/exp(100) saturate near maxpos rather
+::  than tracking the true value; sin(10) blows up to a huge magnitude instead
+::  of staying in [-1, 1] (no range reduction).  log (atanh form) holds up
+::  better -- log(100) ~ 4.605.  Locked as regression, NOT correctness; #18's
+::  Chebyshev rewrite (range reduction) would fix these.
+++  test-exp10   (expect-eq !>(`@`0x7a57.9ded) !>((exp:u sa)))
+++  test-exp50   (expect-eq !>(`@`0x7ffe.1b02) !>((exp:u sb)))
+++  test-exp100  (expect-eq !>(`@`0x7fff.f02b) !>((exp:u sc)))
+++  test-sin10   (expect-eq !>(`@`0xc74b.a64a) !>((sin:u sa)))
+++  test-log100  (expect-eq !>(`@`0x50e9.b7f7) !>((log:u sc)))
+--
diff --git a/saloon/desk/lib/saloon.hoon b/saloon/desk/lib/saloon.hoon
index dc4d5f9..cc755cb 100644
--- a/saloon/desk/lib/saloon.hoon
+++ b/saloon/desk/lib/saloon.hoon
@@ -405,6 +405,16 @@
         %4  ?-(fun %neg neg:rph:unum, %factorial factorial:rph:unum, %exp exp:rph:unum, %sin sin:rph:unum, %cos cos:rph:unum, %tan tan:rph:unum, %log log:rph:unum, %log-10 log-10:rph:unum, %log-2 log-2:rph:unum, %sqrt sqt:rph:unum, %cbrt cbrt:rph:unum)
         %3  ?-(fun %neg neg:rpb:unum, %factorial factorial:rpb:unum, %exp exp:rpb:unum, %sin sin:rpb:unum, %cos cos:rpb:unum, %tan tan:rpb:unum, %log log:rpb:unum, %log-10 log-10:rpb:unum, %log-2 log-2:rpb:unum, %sqrt sqt:rpb:unum, %cbrt cbrt:rpb:unum)
       ==
+      ::
+      ::  complex (/lib/complex): bloq 5/6/7/8 = ch/cs/cd/cq.  Real-only / ordered
+      ::  functions (factorial, log-10, log-2, cbrt) are undefined on complex.
+        %cplx
+      ?+    bloq  !!
+        %8  ?-(fun %neg ~(neg cq:complex rnd), %exp ~(cexp cq:complex rnd), %sin ~(csin cq:complex rnd), %cos ~(ccos cq:complex rnd), %tan ~(ctan cq:complex rnd), %log ~(clog cq:complex rnd), %sqrt ~(csqrt cq:complex rnd), %factorial |=(@ !!), %log-10 |=(@ !!), %log-2 |=(@ !!), %cbrt |=(@ !!))
+        %7  ?-(fun %neg ~(neg cd:complex rnd), %exp ~(cexp cd:complex rnd), %sin ~(csin cd:complex rnd), %cos ~(ccos cd:complex rnd), %tan ~(ctan cd:complex rnd), %log ~(clog cd:complex rnd), %sqrt ~(csqrt cd:complex rnd), %factorial |=(@ !!), %log-10 |=(@ !!), %log-2 |=(@ !!), %cbrt |=(@ !!))
+        %6  ?-(fun %neg ~(neg cs:complex rnd), %exp ~(cexp cs:complex rnd), %sin ~(csin cs:complex rnd), %cos ~(ccos cs:complex rnd), %tan ~(ctan cs:complex rnd), %log ~(clog cs:complex rnd), %sqrt ~(csqrt cs:complex rnd), %factorial |=(@ !!), %log-10 |=(@ !!), %log-2 |=(@ !!), %cbrt |=(@ !!))
+        %5  ?-(fun %neg ~(neg ch:complex rnd), %exp ~(cexp ch:complex rnd), %sin ~(csin ch:complex rnd), %cos ~(ccos ch:complex rnd), %tan ~(ctan ch:complex rnd), %log ~(clog ch:complex rnd), %sqrt ~(csqrt ch:complex rnd), %factorial |=(@ !!), %log-10 |=(@ !!), %log-2 |=(@ !!), %cbrt |=(@ !!))
+      ==
     ==  ::  kind
   ::
   +$  binary-ops  $?  %pow-n
@@ -453,6 +463,14 @@
         %4  ?-(fun %pow-n pow-n:rph:unum, %pow pow:rph:unum)
         %3  ?-(fun %pow-n pow-n:rpb:unum, %pow pow:rpb:unum)
       ==
+      ::  complex: z^w via +cpow (both rays complex); integer +pow-n undefined.
+        %cplx
+      ?+    bloq.meta  !!
+        %8  ?-(fun %pow ~(cpow cq:complex rnd), %pow-n |=([@ @] !!))
+        %7  ?-(fun %pow ~(cpow cd:complex rnd), %pow-n |=([@ @] !!))
+        %6  ?-(fun %pow ~(cpow cs:complex rnd), %pow-n |=([@ @] !!))
+        %5  ?-(fun %pow ~(cpow ch:complex rnd), %pow-n |=([@ @] !!))
+      ==
     ==  ::  kind
   ::
   +|  %linalg
diff --git a/saloon/desk/tests/lib/saloon-cplx.hoon b/saloon/desk/tests/lib/saloon-cplx.hoon
new file mode 100644
index 0000000..4def9fa
--- /dev/null
+++ b/saloon/desk/tests/lib/saloon-cplx.hoon
@@ -0,0 +1,27 @@
+::  Saloon Tier-B transcendentals over %cplx (complex) rays.  Verifies the
+::  trans-scalar/fun-scalar %cplx dispatch reaches /lib/complex per width.
+::  Expected values are the same series outputs checked in /tests/lib/complex-fns
+::  (validated vs numpy).  Inputs: z = 1+2i at @cs (bloq 6) and @ch (bloq 5).
+::
+/-  ls=lagoon
+/+  *test, *saloon, *lagoon
+|%
+::  a 1-element %cplx ray at bloq .b holding packed complex pattern .p
+++  ur  |=([b=@ p=@] ^-(ray:ls (en-ray:(lake %n) [[~[1] b %cplx ~] ~[p]])))
+++  hd   |=(a=ray:ls ^-(@ -:(ravel:(lake %n) a)))
+++  sad  (sake %n .1e-5)   :: %n rounding (bare +sa default rnd is not %n)
+++  z6   `@`0x4000.0000.3f80.0000               ::  1+2i @cs
+++  w26  `@`0x4000.0000                          ::  2+0i @cs
+++  z5   `@`0x4000.3c00                          ::  1+2i @ch
+++  test-cplx-exp-cs   (expect-eq !>(`@`0x401e.30c5.bf90.cb4e) !>((hd (exp:sad (ur 6 z6)))))
+++  test-cplx-log-cs   (expect-eq !>(`@`0x3f8d.b70a.3f4e.0210) !>((hd (log:sad (ur 6 z6)))))
+++  test-cplx-sqrt-cs  (expect-eq !>(`@`0x3f49.4138.3fa2.d18a) !>((hd (sqrt:sad (ur 6 z6)))))
+++  test-cplx-sin-cs   (expect-eq !>(`@`0x3ffa.d435.404a.9c1e) !>((hd (sin:sad (ur 6 z6)))))
+++  test-cplx-cos-cs   (expect-eq !>(`@`0xc043.524b.4002.1823) !>((hd (cos:sad (ur 6 z6)))))
+++  test-cplx-tan-cs   (expect-eq !>(`@`0x3f81.e4c1.3d0a.7f5c) !>((hd (tan:sad (ur 6 z6)))))
+++  test-cplx-pow-cs   (expect-eq !>(`@`0x4080.0004.c03f.fff5) !>((hd (pow:sad (ur 6 z6) (ur 6 w26)))))
+++  test-cplx-exp-ch   (expect-eq !>(`@`0x40f1.bc86) !>((hd (exp:sad (ur 5 z5)))))
+::  real-only / ordered functions are undefined on complex -> crash.
+++  test-cplx-factorial-crashes  (expect-fail |.((factorial:sad (ur 6 z6))))
+++  test-cplx-cbrt-crashes       (expect-fail |.((cbrt:sad (ur 6 z6))))
+--