Taylor Series, Integrals, Differentials and parts, see also identities.txt macsyma This is Macsyma 412.61 Copyright (c) 1982 Massachusetts Institute of Technology. TAYLOR SERIES: taylor(sqrt(1+x),x,0,9); 2 3 4 5 6 7 8 9 X X X 5 X 7 X 21 X 33 X 429 X 715 X 1 + - - -- + -- - ---- + ---- - ----- + ----- - ------ + ------ + . . . 2 8 16 128 256 1024 2048 32768 65536 taylor(sqrt(x),x,0,9); SQRT(X) + . . . taylor(sqrt(1/x),x,0,4); 1 ------- + . . . SQRT(X) taylor(exp(x),x,0,7); 2 3 4 5 6 7 X X X X X X 1 + X + -- + -- + -- + --- + --- + ---- + . . . 2 6 24 120 720 5040 taylor(log(1+x),x,0,7); 2 3 4 5 6 7 X X X X X X X - -- + -- - -- + -- - -- + -- + . . . 2 3 4 5 6 7 taylor(log(x),x,1,9); 2 3 4 5 6 7 (X - 1) (X - 1) (X - 1) (X - 1) (X - 1) (X - 1) X - 1 - -------- + -------- - -------- + -------- - -------- + -------- 2 3 4 5 6 7 taylor(log(x),x,0,9); LOG(X) + . . . taylor(log(1/x),x,0,4); - LOG(X) + . . . taylor(sin(x),x,0,9); 3 5 7 9 X X X X X - -- + --- - ---- + ------ + . . . 6 120 5040 362880 taylor(cos(x),x,0,9); 2 4 6 8 X X X X 1 - -- + -- - --- + ----- + . . . 2 24 720 40320 taylor(tan(x),x,0,9); 3 5 7 9 X 2 X 17 X 62 X X + -- + ---- + ----- + ----- + . . . 3 15 315 2835 taylor(tan(x+%pi/2),x,0,5); 3 5 1 X X 2 X - - + - + -- + ---- + . . . X 3 45 945 taylor(tan(%i*y+%pi/2),y,0,5); 3 5 %I %I Y %I Y (2 %I) Y -- + ---- - ----- + --------- + . . . Y 3 45 945 taylor(cot(x),x,0,9); 3 5 7 9 1 X X 2 X X 2 X - - - - -- - ---- - ---- - ----- + . . . X 3 45 945 4725 93555 taylor(cot(%PI/2+x),x,0,9); 3 5 7 9 X 2 X 17 X 62 X - X - -- - ---- - ----- - ----- + . . . 3 15 315 2835 taylor(cot(%i*y),y,0,5); 3 5 %I %I Y %I Y (2 %I) Y - -- - ---- + ----- - --------- + . . . Y 3 45 945 taylor(cot(x+%pi),x,0,5); 3 5 1 X X 2 X - - - - -- - ---- + . . . X 3 45 945 taylor(cot(%i*y+%pi),y,0,5); 3 5 %I %I Y %I Y (2 %I) Y - -- - ---- + ----- - --------- + . . . Y 3 45 945 taylor(cot(1/x),x,0,5); TAYLOR encountered an essential singularity in above taylor(asin(x),x,0,9); 3 5 7 9 X 3 X 5 X 35 X X + -- + ---- + ---- + ----- + . . . 6 40 112 1152 taylor(asin(1-x),x,0,9); 3/2 5/2 %PI SQRT(2) X (3 SQRT(2)) X --- - SQRT(2) SQRT(X) - ------------ - ---------------- 2 12 160 taylor(asin(x+1),x,0,6); 3/2 5/2 %PI (SQRT(2) %I) X (3 SQRT(2) %I) X --- - SQRT(2) %I SQRT(X) + ----------------- - ------------------- 2 12 160 7/2 9/2 11/2 (5 SQRT(2) %I) X (35 SQRT(2) %I) X (63 SQRT(2) %I) X + ------------------- - -------------------- + --------------------- + . . 896 18432 90112 taylor(asin(x-1),x,0,9); 3/2 5/2 %PI SQRT(2) X (3 SQRT(2)) X - --- + SQRT(2) SQRT(X) + ------------ + ---------------- 2 12 160 taylor(asin(1+%i*y),y,0,3); %PI 1/4 --- - (- 1) SQRT(2) %I SQRT(Y) 2 1/4 3 1/4 3/2 (8 ((- 1) ) SQRT(2) %I + 9 (- 1) SQRT(2)) Y - ---------------------------------------------------- 12 1/4 1/4 3 5/2 (243 (- 1) SQRT(2) %I - 240 ((- 1) ) SQRT(2)) Y + -------------------------------------------------------- 160 taylor(asin(%i*y-1),y,0,3); %PI 1/4 - --- + (- 1) SQRT(2) SQRT(Y) 2 1/4 1/4 3 3/2 (9 (- 1) SQRT(2) %I - 8 ((- 1) ) SQRT(2)) Y + ---------------------------------------------------- 12 1/4 3 1/4 5/2 (240 ((- 1) ) SQRT(2) %I + 243 (- 1) SQRT(2)) Y - -------------------------------------------------------- 160 taylor(asin(1/x),x,0,4); 2 4 2 %I %I X (3 %I) X - LOG(----) %I + ----- + --------- + . . . X 4 32 taylor(acos(x),x,0,9); 3 5 7 9 %PI X 3 X 5 X 35 X --- - X - -- - ---- - ---- - ----- + . . . 2 6 40 112 1152 taylor(acos(1+x),x,0,9); 3/2 5/2 (SQRT(2) %I) X (3 SQRT(2) %I) X SQRT(2) %I SQRT(X) - ----------------- + ------------------- 12 160 7/2 9/2 11/2 (5 SQRT(2) %I) X (35 SQRT(2) %I) X (63 SQRT(2) %I) X - ------------------- + -------------------- - --------------------- 896 18432 90112 taylor(acos(x-1),x,0,9); 3/2 5/2 SQRT(2) X (3 SQRT(2)) X %PI - SQRT(2) SQRT(X) - ------------ - ---------------- 12 160 7/2 9/2 11/2 (5 SQRT(2)) X (35 SQRT(2)) X (63 SQRT(2)) X - ---------------- - ----------------- - ------------------ 896 18432 90112 taylor(acos(1+%i*y),y,0,3); 1/4 (- 1) SQRT(2) %I SQRT(Y) 1/4 3 1/4 3/2 (8 ((- 1) ) SQRT(2) %I + 9 (- 1) SQRT(2)) Y + ---------------------------------------------------- 12 1/4 1/4 3 5/2 (243 (- 1) SQRT(2) %I - 240 ((- 1) ) SQRT(2)) Y - -------------------------------------------------------- 160 taylor(acos(%i+x),x,0,9); 2 3 %PI + 2 LOG(SQRT(2) - 1) %I SQRT(2) X (SQRT(2) %I) X SQRT(2) X --------------------------- - --------- - --------------- + ---------- 2 2 8 48 taylor(acos(-%i+x),x,0,9); 2 3 %PI + 2 LOG(SQRT(2) + 1) %I SQRT(2) X (SQRT(2) %I) X SQRT(2) X --------------------------- - --------- + --------------- + ---------- 2 2 8 48 taylor(acos(%i*y-1),y,0,3); 1/4 %PI - (- 1) SQRT(2) SQRT(Y) 1/4 1/4 3 3/2 (9 (- 1) SQRT(2) %I - 8 ((- 1) ) SQRT(2)) Y - ---------------------------------------------------- 12 1/4 3 1/4 5/2 (240 ((- 1) ) SQRT(2) %I + 243 (- 1) SQRT(2)) Y + -------------------------------------------------------- + . . . 160 taylor(acos(1/x),x,0,4); 2 %I %PI + 2 LOG(----) %I 2 4 X %I X (3 %I) X -------------------- - ----- - --------- + . . . 2 4 32 taylor(atan(x),x,0,9); 3 5 7 9 X X X X X - -- + -- - -- + -- + . . . 3 5 7 9 taylor(atan(x+1),x,0,9); 2 3 5 6 7 9 %PI X X X X X X X --- + - - -- + -- - -- + -- - --- + --- + . . . 4 2 4 12 40 48 112 288 taylor(atan(x-1),x,0,6); 2 3 5 6 %PI X X X X X - --- + - + -- + -- - -- - -- + . . . 4 2 4 12 40 48 taylor(atan(%i+x),x,0,5); 2 %I LOG(- ----) %I 2 3 4 5 X X %I X X %I X X -------------- + - + ----- - -- - ----- + --- + . . . 2 4 16 48 128 320 taylor(atan(-%i+x),x,0,5); %I X LOG(- ----) %I 2 3 4 5 2 X %I X X %I X X -------------- + - - ----- - -- + ----- + --- + . . . 2 4 16 48 128 320 taylor(atan(%i+%i*y),y,0,5); 2 LOG(- -) %I 2 3 4 5 Y %I Y %I Y %I Y %I Y %I Y ----------- + ---- - ----- + ----- - ----- + ----- + . . . 2 4 16 48 128 320 taylor(atan(-%i+%i*y),y,0,5); Y LOG(-) %I 2 3 4 5 2 %I Y %I Y %I Y %I Y %I Y --------- + ---- + ----- + ----- + ----- + ----- + . . . 2 4 16 48 128 320 taylor(acot(x),x,0,9); Is X positive or negative? positive; 3 5 7 9 %PI X X X X --- - X + -- - -- + -- - -- + . . . 2 3 5 7 9 taylor(acot(x),x,0,9); Is X positive or negative? negative; 3 5 7 9 %PI X X X X - --- - X + -- - -- + -- - -- + . . . 2 3 5 7 9 taylor(acot(%i+x),x,0,9); %I X LOG(- ----) %I 2 3 4 5 6 7 8 2 X %I X X %I X X %I X X %I X -------------- - - - ----- + -- + ----- - --- - ----- + ---- + ----- 2 4 16 48 128 320 768 1792 4096 taylor(acot(-%i+x),x,0,9); 2 %I LOG(- ----) %I 2 3 4 5 6 7 8 X X %I X X %I X X %I X X %I X -------------- - - + ----- + -- - ----- - --- + ----- + ---- - ----- 2 4 16 48 128 320 768 1792 4096 taylor(acot(-%i+%i*y),y,0,5); 2 LOG(- -) %I 2 3 4 5 Y %I Y %I Y %I Y %I Y %I Y ----------- - ---- - ----- - ----- - ----- - ----- + . . . 2 4 16 48 128 320 taylor(acot(1+x),x,0,9); 2 3 5 6 7 9 %PI X X X X X X X --- - - + -- - -- + -- - -- + --- - --- + . . . 4 2 4 12 40 48 112 288 taylor(acot(x-1),x,0,9); 2 3 5 6 7 9 %PI X X X X X X X - --- - - - -- - -- + -- + -- + --- - --- + . . . 4 2 4 12 40 48 112 288 taylor(acot(%i+%i*y),y,0,5); Y LOG(-) %I 2 3 4 5 2 %I Y %I Y %I Y %I Y %I Y --------- - ---- + ----- - ----- + ----- - ----- + . . . 2 4 16 48 128 320 taylor(acot(1/x),x,0,5); 3 5 X X X - -- + -- + . . . 3 5 taylor(sinh(x),x,0,9); 3 5 7 9 X X X X X + -- + --- + ---- + ------ + . . . 6 120 5040 362880 taylor(cosh(x),x,0,9); 2 4 6 8 X X X X 1 + -- + -- + --- + ----- + . . . 2 24 720 40320 taylor(tanh(x),x,0,9); 3 5 7 9 X 2 X 17 X 62 X X - -- + ---- - ----- + ----- + . . . 3 15 315 2835 taylor(coth(x),x,0,9); 3 5 7 9 1 X X 2 X X 2 X - + - - -- + ---- - ---- + ----- + . . . X 3 45 945 4725 93555 taylor(coth(1/x),x,0,5); TAYLOR encountered an essential singularity in above taylor(asinh(x),x,0,9); 3 5 7 9 X 3 X 5 X 35 X X - -- + ---- - ---- + ----- + . . . 6 40 112 1152 taylor(asinh(%i+x),x,0,3); %PI %I 1/4 ------ - (- 1) SQRT(2) %I SQRT(X) 2 1/4 3 1/4 3/2 (8 ((- 1) ) SQRT(2) %I + 9 (- 1) SQRT(2)) X + ---------------------------------------------------- 12 1/4 1/4 3 5/2 (243 (- 1) SQRT(2) %I - 240 ((- 1) ) SQRT(2)) X + -------------------------------------------------------- 160 taylor(asinh(%i+%i*y),y,0,3); 3/2 5/2 %PI %I SQRT(2) Y (3 SQRT(2)) Y ------ + SQRT(2) SQRT(Y) - ------------ + ---------------- + . . . 2 12 160 taylor(asinh(-%i+x),x,0,3); %PI %I 1/4 - ------ - (- 1) SQRT(2) SQRT(X) 2 1/4 1/4 3 3/2 (9 (- 1) SQRT(2) %I - 8 ((- 1) ) SQRT(2)) X + ---------------------------------------------------- 12 1/4 3 1/4 5/2 (240 ((- 1) ) SQRT(2) %I + 243 (- 1) SQRT(2)) X + -------------------------------------------------------- + . . . 160 taylor(asinh(-%i+%i*y),y,0,3); 3/2 5/2 %PI %I (SQRT(2) %I) Y (3 SQRT(2) %I) Y - ------ + SQRT(2) %I SQRT(Y) + ----------------- + ------------------- 2 12 160 taylor(asinh(1+x),x,0,3); 2 3 SQRT(2) X SQRT(2) X SQRT(2) X LOG(SQRT(2) + 1) + --------- - ---------- + ---------- + . . . 2 8 48 taylor(asinh(x-1),x,0,3); 2 3 SQRT(2) X SQRT(2) X SQRT(2) X LOG(SQRT(2) - 1) + --------- + ---------- + ---------- + . . . 2 8 48 taylor(asinh(1+%i*y),y,0,3); 2 3 (SQRT(2) %I) Y SQRT(2) Y (SQRT(2) %I) Y LOG(SQRT(2) + 1) + -------------- + ---------- - --------------- 2 8 48 taylor(asinh(%i*y-1),y,0,3); 2 3 (SQRT(2) %I) Y SQRT(2) Y (SQRT(2) %I) Y LOG(SQRT(2) - 1) + -------------- - ---------- - --------------- 2 8 48 taylor(asinh(1/x),x,0,4); 2 4 2 X 3 X LOG(-) + -- - ---- + . . . X 4 32 taylor(acosh(x),x,0,9); 3 5 7 9 %I %PI %I X (3 %I) X (5 %I) X (35 %I) X - ------ + %I X + ----- + --------- + --------- + ---------- + . . . 2 6 40 112 1152 taylor(acosh(%i+x),x,0,3); 2 %I %PI - 2 LOG(SQRT(2) - 1) (SQRT(2) %I) X SQRT(2) X - --------------------------- + -------------- - ---------- 2 2 8 taylor(acosh(%i+%i*y),y,0,3); 2 3 %I %PI - 2 LOG(SQRT(2) - 1) SQRT(2) Y SQRT(2) Y SQRT(2) Y - --------------------------- - --------- + ---------- - ---------- 2 2 8 48 taylor(acosh(-%i+x),x,0,3); 2 %I %PI - 2 LOG(SQRT(2) + 1) (SQRT(2) %I) X SQRT(2) X - --------------------------- + -------------- + ---------- 2 2 8 taylor(acosh(-%i+%i*y),y,0,3); 2 3 %I %PI - 2 LOG(SQRT(2) + 1) SQRT(2) Y SQRT(2) Y SQRT(2) Y - --------------------------- - --------- - ---------- - ---------- 2 2 8 48 taylor(acosh(1+x),x,0,3); 3/2 5/2 SQRT(2) X (3 SQRT(2)) X SQRT(2) SQRT(X) - ------------ + ---------------- + . . . 12 160 taylor(acosh(x-1),x,0,3); 3/2 5/2 (%I SQRT(2)) X (3 %I SQRT(2)) X - %I %PI + %I SQRT(2) SQRT(X) + ----------------- + ------------------- 12 160 taylor(acosh(1+%i*y),y,0,3); 1/4 (- 1) SQRT(2) SQRT(Y) 1/4 1/4 3 3/2 (9 (- 1) SQRT(2) %I - 8 ((- 1) ) SQRT(2)) Y - ---------------------------------------------------- 12 1/4 3 1/4 5/2 (240 ((- 1) ) SQRT(2) %I + 243 (- 1) SQRT(2)) Y - -------------------------------------------------------- + . . . 160 taylor(acosh(1/x),x,0,4); 2 %I %I %PI - 2 LOG(----) 2 4 X X 3 X - -------------------- - -- - ---- + . . . 2 4 32 taylor(atanh(x),x,0,9); 3 5 7 9 X X X X X + -- + -- + -- + -- + . . . 3 5 7 9 taylor(atanh(1+x),x,0,5); 2 LOG(- -) 2 3 4 5 X X X X X X -------- + - - -- + -- - --- + --- + . . . 2 4 16 48 128 320 taylor(atanh(x-1),x,0,5); X LOG(-) 2 3 4 5 2 X X X X X ------ + - + -- + -- + --- + --- + . . . 2 4 16 48 128 320 taylor(atanh(1+%i*y),y,0,5); 2 %I LOG(----) 2 3 4 5 Y %I Y Y %I Y Y %I Y --------- + ---- + -- - ----- - --- + ----- + . . . 2 4 16 48 128 320 taylor(atanh(%i*y-1),y,0,5); %I Y LOG(----) 2 3 4 5 2 %I Y Y %I Y Y %I Y --------- + ---- - -- - ----- + --- + ----- + . . . 2 4 16 48 128 320 taylor(acoth(x),x,0,9); 3 5 7 9 %PI %I X X X X ------ + X + -- + -- + -- + -- + . . . 2 3 5 7 9 taylor(acoth(1+x),x,0,9); 2 LOG(-) 2 3 4 5 6 7 8 9 X X X X X X X X X X ------ + - - -- + -- - --- + --- - --- + ---- - ---- + ---- + . . . 2 4 16 48 128 320 768 1792 4096 9216 taylor(acoth(x-1),x,0,9); X LOG(- -) 2 3 4 5 6 7 8 9 2 X X X X X X X X X -------- + - + -- + -- + --- + --- + --- + ---- + ---- + ---- + . . . 2 4 16 48 128 320 768 1792 4096 9216 taylor(acoth(%i+x),x,0,9); 2 3 5 6 7 9 %PI %I X %I X X X %I X X X - ------ + - + ----- - -- - -- - ----- + --- + --- + . . . 4 2 4 12 40 48 112 288 taylor(acoth(-%i+x),x,0,9); 2 3 5 6 7 9 %PI %I X %I X X X %I X X X ------ + - - ----- - -- - -- + ----- + --- + --- + . . . 4 2 4 12 40 48 112 288 taylor(acoth(1+%i*y),y,0,5); 2 %I LOG(- ----) 2 3 4 5 Y %I Y Y %I Y Y %I Y ----------- + ---- + -- - ----- - --- + ----- + . . . 2 4 16 48 128 320 taylor(acoth(%i*y-1),y,0,5); %I Y LOG(- ----) 2 3 4 5 2 %I Y Y %I Y Y %I Y ----------- + ---- - -- - ----- + --- + ----- + . . . 2 4 16 48 128 320 taylor(acoth(1/x),x,0,5); 3 5 X X X + -- + -- + . . . 3 5 INDEFINITE INTEGRALS: integrate(sqrt(x),x); 3/2 2 X ------ 3 integrate(log(x),x); X LOG(X) - X integrate(exp(x),x); X %E integrate(sin(x),x); - COS(X) integrate(cos(x),x); SIN(X) integrate(tan(x),x); LOG(SEC(X)) integrate(cot(x),x); LOG(SIN(X)) integrate(asin(x),x); 2 X ASIN(X) + SQRT(1 - X ) integrate(acos(x),x); 2 X ACOS(X) - SQRT(1 - X ) integrate(atan(x),x); 2 LOG(X + 1) X ATAN(X) - ----------- 2 integrate(acot(x),x); 2 LOG(X + 1) ----------- + X ACOT(X) 2 integrate(sinh(x),x); COSH(X) integrate(cosh(x),x); SINH(X) integrate(tanh(x),x); LOG(COSH(X)) integrate(coth(x),x); LOG(SINH(X)) integrate(asinh(x),x); 2 X ASINH(X) - SQRT(X + 1) integrate(acosh(x),x); 2 X ACOSH(X) - SQRT(X - 1) integrate(atanh(x),x); 2 LOG(1 - X ) ----------- + X ATANH(X) 2 integrate(acoth(x),x); 2 LOG(X - 1) ----------- + X ACOTH(X) 2 DIFFERENTIALS: diff(sqrt(x),x); 1 --------- 2 SQRT(X) diff(log(x),x); 1 - X diff(exp(x),x); X %E diff(sin(x),x); COS(X) diff(cos(x),x); - SIN(X) diff(tan(x),x); 2 SEC (X) diff(cot(x),x); 2 - CSC (X) diff(asin(x),x); 1 ------------ 2 SQRT(1 - X ) diff(acos(x),x); 1 - ------------ 2 SQRT(1 - X ) diff(atan(x),x); 1 ------ 2 X + 1 diff(acot(x),x); 1 - ------ 2 X + 1 diff(sinh(x),x); COSH(X) diff(cosh(x),x); SINH(X) diff(tanh(x),x); 2 SECH (X) diff(coth(x),x); 2 - CSCH (X) diff(asinh(x),x); 1 ------------ 2 SQRT(X + 1) diff(acosh(x),x); 1 ------------ 2 SQRT(X - 1) diff(atanh(x),x); 1 ------ 2 1 - X diff(acoth(x),x); 1 - ------ 2 X - 1 Partial Differentials for real and imaginary parts: diff(sqrt(x+%i*y),x); 1 ---------------- 2 SQRT(%I Y + X) diff(sqrt(x+%i*y),y); %I ---------------- 2 SQRT(%I Y + X) diff(log(x+%i*y),x); 1 -------- %I Y + X diff(log(x+%i*y),y); %I -------- %I Y + X diff(exp(x+%i*y),x); %I Y + X %E diff(exp(x+%i*y),y); %I Y + X %I %E diff(sin(x+%i*y),x); COS(%I Y + X) diff(sin(x+%i*y),y); %I COS(%I Y + X) diff(cos(x+%i*y),x); - SIN(%I Y + X) diff(cos(x+%i*y),y); - %I SIN(%I Y + X) diff(asin(x+%i*y),x); 1 --------------------- 2 SQRT(1 - (%I Y + X) ) diff(asin(x+%i*y),y); %I --------------------- 2 SQRT(1 - (%I Y + X) ) diff(acoth(x+%i*y),x); 1 - --------------- 2 (%I Y + X) - 1 diff(acoth(x+%i*y),y); %I - --------------- 2 (%I Y + X) - 1 quit(); Maple finds reasonably simple expressions for arctan(z), arccot(z), arctanh(z), and arccoth(z) in the form u+iv, where u and v are real. Here they are: 1 1 x^2 + (y+1)^2 arctan(x+iy) = - (arctan(x,1-y) - arctan(-x,1+y)) + - i log ------------- 2 4 x^2 + (y-1)^2 1 1 x^2 + (y+1)^2 arccot(x+iy) = - (Pi - arctan(x,1-y) + arctan(-x,1+y)) - - i log ------------- 2 4 x^2 + (y-1)^2 1 (x+1)^2 + y^2 1 arctanh(x+iy) = - log ------------- + - i (arctan(y,1+x) - arctan(-y,1-x)) 4 (x-1)^2 + y^2 2 1 (x+1)^2 + y^2 1 arccoth(x+iy) = - log ------------- + - i (Pi + arctan(y,1+x) - arctan(-y,1-x)) 4 (x-1)^2 + y^2 2 It also finds similar expressions for arcsin(z), arccos(z), arcsinh(z), and arccosh(z), but they are not simple enough to be attractive. The real functions involved are log, arcsin, arccos, and sqrt. Real and Imaginary parts of arc functions from real functions and arithmetic (from MACSYMA, I am not sure about the question about positive or negative) realpart(asin(x+%i*y)); Is X Y positive, negative, or zero? positive; 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) - Y + X - 1) - ATAN2(-------------------------------------------------- - X, SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) + Y - X + 1) -------------------------------------------------- - Y) SQRT(2) imagpart(asin(x+%i*y)); Is X Y positive, negative, or zero? positive; 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) + Y - X + 1) 2 - LOG((-------------------------------------------------- - Y) SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) - Y + X - 1) 2 + (X - --------------------------------------------------) )/2 SQRT(2) realpart(asin(x+%i*y)); Is X Y positive, negative, or zero? negative; 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) - Y + X - 1) ATAN2(-------------------------------------------------- + X, SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) + Y - X + 1) -------------------------------------------------- - Y) SQRT(2) imagpart(asin(x+%i*y)); Is X Y positive, negative, or zero? negative; 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) + Y - X + 1) 2 - LOG((-------------------------------------------------- - Y) SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) - Y + X - 1) 2 + (-------------------------------------------------- + X) )/2 SQRT(2) realpart(acos(x+%i*y)); Is X Y positive, negative, or zero? positive; 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) + Y - X + 1) ATAN2(-------------------------------------------------- + Y, SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) - Y + X - 1) -------------------------------------------------- + X) SQRT(2) imagpart(acos(x+%i*y)); Is X Y positive, negative, or zero? positive; 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) + Y - X + 1) 2 - LOG((-------------------------------------------------- + Y) SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) - Y + X - 1) 2 + (-------------------------------------------------- + X) )/2 SQRT(2) realpart(acos(x+%i*y)); Is X Y positive, negative, or zero? negative; 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) + Y - X + 1) ATAN2(-------------------------------------------------- + Y, SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) - Y + X - 1) X - --------------------------------------------------) SQRT(2) imagpart(acos(x+%i*Y)); Is X Y positive, negative, or zero? negative; 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) + Y - X + 1) 2 - LOG((-------------------------------------------------- + Y) SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((Y - X + 1) + 4 X Y ) - Y + X - 1) 2 + (X - --------------------------------------------------) )/2 SQRT(2) realpart(atan(x+%i*y)); X Y X (1 - Y) X ATAN2(------------- + ------------- + -------------, 2 2 2 2 2 2 (1 - Y) + X (1 - Y) + X (1 - Y) + X 2 (1 - Y) Y 1 - Y X ------------- + ------------- - -------------)/2 2 2 2 2 (1 - Y) + X (1 - Y) + X (1 - Y) + X factor(%); this becomes::::::::::::::::::::::::::: 2 2 2 X Y + X - 1 ATAN2(-----------------, - -----------------) /2 2 2 2 2 Y - 2 Y + X + 1 Y - 2 Y + X + 1 imagpart(atan(x+%i*y)); 2 2 2 Y X 1 2 4 X LOG((- ------------- - ------------- + -------------) + ----------------)/4 2 2 2 2 2 2 2 2 2 (1 - Y) + X (1 - Y) + X (1 - Y) + X ((1 - Y) + X ) factor(%); this becomes::::::::::: 2 2 Y + 2 Y + X + 1 LOG(-----------------) /4 2 2 Y - 2 Y + X + 1 realpart(acot(x+%i*y)); X Y X (- Y - 1) X - ATAN2(--------------- + --------------- - ---------------, 2 2 2 2 2 2 (- Y - 1) + X (- Y - 1) + X (- Y - 1) + X 2 (- Y - 1) Y - Y - 1 X - --------------- + --------------- + ---------------)/2 2 2 2 2 2 2 (- Y - 1) + X (- Y - 1) + X (- Y - 1) + X imagpart(acot(x+%i*y)); 2 2 Y X 1 2 LOG((--------------- + --------------- - ---------------) 2 2 2 2 2 2 (- Y - 1) + X (- Y - 1) + X (- Y - 1) + X 2 4 X + ------------------)/4 2 2 2 ((- Y - 1) + X ) realpart(asinh(x+%i*y)); Is X Y positive, negative, or zero? positive; 2 2 2 2 2 2 2 SQRT(SQRT((- Y + X + 1) + 4 X Y ) + Y - X - 1) 2 LOG((---------------------------------------------------- + Y) SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((- Y + X + 1) + 4 X Y ) - Y + X + 1) 2 + (---------------------------------------------------- + X) )/2 SQRT(2) imagpart(asinh(x+%i*y)); Is X Y positive, negative, or zero? positive; 2 2 2 2 2 2 2 SQRT(SQRT((- Y + X + 1) + 4 X Y ) + Y - X - 1) ATAN2(---------------------------------------------------- + Y, SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((- Y + X + 1) + 4 X Y ) - Y + X + 1) ---------------------------------------------------- + X) SQRT(2) realpart(acosh(x+%i*y)); Is X Y positive, negative, or zero? positive; 2 2 2 2 2 2 2 SQRT(SQRT((- Y + X - 1) + 4 X Y ) + Y - X + 1) 2 LOG((---------------------------------------------------- + Y) SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((- Y + X - 1) + 4 X Y ) - Y + X - 1) 2 + (---------------------------------------------------- + X) )/2 SQRT(2) imagpart(acosh(x+%i*y)); Is X Y positive, negative, or zero? positive; 2 2 2 2 2 2 2 SQRT(SQRT((- Y + X - 1) + 4 X Y ) + Y - X + 1) ATAN2(---------------------------------------------------- + Y, SQRT(2) 2 2 2 2 2 2 2 SQRT(SQRT((- Y + X - 1) + 4 X Y ) - Y + X - 1) ---------------------------------------------------- + X) SQRT(2) realpart(atanh(x+%i*y)); 2 2 2 Y X 1 2 4 Y LOG((- ------------- - ------------- + -------------) + ----------------)/4 2 2 2 2 2 2 2 2 2 Y + (1 - X) Y + (1 - X) Y + (1 - X) (Y + (1 - X) ) imagpart(atanh(x+%i*y)); X Y (1 - X) Y Y ATAN2(------------- + ------------- + -------------, 2 2 2 2 2 2 Y + (1 - X) Y + (1 - X) Y + (1 - X) 2 Y (1 - X) X 1 - X - ------------- + ------------- + -------------)/2 2 2 2 2 2 2 Y + (1 - X) Y + (1 - X) Y + (1 - X) realpart(acoth(x+%i*y)); 2 2 2 Y X 1 2 4 Y LOG((------------- + ------------- - -------------) + ----------------)/4 2 2 2 2 2 2 2 2 2 Y + (X - 1) Y + (X - 1) Y + (X - 1) (Y + (X - 1) ) imagpart(acoth(x+%i*y)); X Y (X - 1) Y Y - ATAN2(------------- - ------------- + -------------, 2 2 2 2 2 2 Y + (X - 1) Y + (X - 1) Y + (X - 1) 2 Y (X - 1) X X - 1 ------------- + ------------- + -------------)/2 2 2 2 2 2 2 Y + (X - 1) Y + (X - 1) Y + (X - 1)