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6 II 3 6. π 3.459... ( /( π 33 π 00 π 34 6.. ( (a cos π 2 0 π (0, 2 3 π (b z C, m, Z ( ( cos z + π 2 (, si z + π 2 (cos z, si z, 4m, ( si z, cos z, 4m +, (cos z, si z, 4m + 2, (si z, cos z, 4m + 3. (6. cos, si 2π cos(z + 2π cos z, si(z + 2π si z. (c cos, si [0, 2π] x 0 π/2 π 3π/2 2π cos x 0 0 si x 0 0 0 x R (cos x, si x (, 0 x 2πZ. (a: ( (0, 6 si > 0. a 2 + a 2+ si x ( (2 +! x2+ } {{ } a 2.5.9 (a 2 + a 2+, x4+ (4 +! x4+3 (4 + 3! x4+ (4 +! ( x 2. (4 + 2(4 + 3 33 William Joes (675 749 34 76 J. H. Lambert, 882 C. L. F. Lidema 7

x (0, 6, N x 2 (4 + 2(4 + 3 < 6 2 3, a 2 + a 2+ > 0. x (0, 6 si x > 0. (2 cos [0, 6] (0, 6 (cos si < 0. ( 5.4.6 (2 (3 cos 0, cos 3 < 0. cos 0 cos cos a 2+ + a 2+2 x 2 ( x 2 cos x (2! } {{ } a 2.5.9 (a 2+ + a 2+2 x4+2 (4 + 2! x4+4 (4 + 4! x4+2 (4 + 2! ( x 2 (4 + 3(4 + 4 x 2 (4 + 3(4 + 4 2 3 4, a 2+ + a 2+2 0. x 2 cos x a + a 2 x2 2 ( x2 2 cos 3 3 2 ( 9 4 8, cos 3 /8 < 0. (a (3, cos 2.3.4 c (0, 3, cos c 0. (2 c 2c (b: cos π 2 0 cos2 π 2 + si2 π 2. 0 < π/2 < 3 < 6 ( si π/2 > 0. ( cos π 2, si π 2 (0,. ( 3.2.2 ( cos z + π cos z cos π 2 } {{ 2} 0 si z si π } {{ 2} (, si z + π si z cos π + cos z si π. 2 } {{ 2} } {{ 2} 0 (6. z z π 2 (6. ± (6. Z (6. (c: (6. [0, π/2] [ mπ 2, (m+π 2 ] (m, 2, 3 [0, π/2] cos (2 (0, π/2 si cos > 0. ( 5.4.6 si [0, π/2] 3.2.2 ( 72

6..2 ( (a t, s R e it e is t s 2πZ. (b c R t e it [c, c + 2π S def. {z C ; z } (a: e it e is e i(t s 6.. t s 2πZ. (b: ϕ(t e it (t R e ic ϕ [0, 2π S z e ic z S S c 0 c 0 (a 6.. ϕ(0 ϕ(2π ( ϕ : [0, 2π] S z S, c [0, 2π], z e ic. z S Re z [, ]. cos 0 cos 2π, cos π ( 2.3.4 c + [0, π], c [π, 2π], cos c + cos c Re z. 6.. si c + 0 si c. Im z 0 Im z (Re z 2 cos 2 c + si c +. z Re z + i Im z cos c + + i si c + e ic +. Im z 0 z e ic. ( R (0, 3..3 6..2 R [c, c + 2π (c R C\{0} 6..3 (a z, w C e z e w z w 2πiZ. (b c R z e z {z C ; Im z [c, c + 2π} C\{0} (a e z e w e z w. z w z w 0 : e z ere z 3.4. e z, Re z 0 3..3 x e x R. Re z 0, e z e i Im z. 6..2(a Im z 2πZ. z Re }{{} z +i Im z 2iπZ. 0 6..2(a (b: (a w C\{0} w/ w S 6..2 t [c, c + 2π, w/ w e it. w w e it e log w +it. 73

6.. ( z C ch z 0 z πi 2 +πiz, cos z 0 z π 2 + πz, sh z 0 z πiz, si z 0 z πz. 6..2 f(t (t si t, cos t (t 0 0 < t < π (i f(t C t def. {x R 2 ; x (t, }. (ii C π {x R 2 ; x π, x 2 cos t} g(t f (t, f(π g(t 6..2 f 35 C 0 (0, 0 C 0 x t f(t (ii f(t 638 6..3 ( p N\{0}, ω exp(2πi/p xp (p! p p j0 exp(ωj x p j0 ωj p p, 0. 6..4 ( m N\{0}, f, g : C C, g a C f(a 0, g(a 0, f(z f(a + (z a m g(z (z C a f f(a/g(a re iθ, (r > 0, θ R a f h δ /m e i(θ+π/m (0 < δ r m f(a + g(ah g(a (r δ f(a g(ah m. }{{} } {{ } g(are iθ g(aδe iθ δ g(a + h g(a < g(a. f(a + h f(a + g(ah m + h m (g(a + h g(a < f(a. } {{ } } {{ } f(a g(ah m < g(ah m δ a + h a a f a f 6..4 6..4 ( 6..4 a f 6..5 ( f : C C f(a 0 a C ( lim z f(z, f a C 5.5. a 6..4 f(a 0. 6..5 ( ta (taget : ta z si z cos z, z C\ ( π 2 + πz ( 6.. z cos z 0 (a R\ ( π 2 + πz ta / cos 2 > 0. ta ( π 2, π 2 35 Galileo Galilei (564 642 74

(b lim ta x ±, (. x ± π 2 (a: ta ( si cos cos {}}{ si cos si si {}}{ cos cos 2 cos 2. ta > 0 5.4.6 ( π 2, π 2 (b: 6.. 6..6 x, y, x + y C\( π 2 + πz ta x ta y, ta(x + y ta x+ta y ta x ta y 6..7 a > 0, f(t e at (cos t, si t (t R a ta θ θ (0, π 2 f f f f cos θ f ( θ 6..8 th (hyperbolic taget : th z sh z ( πi ch z, z C\ 2 + πiz ( 6.. z ch z 0 (i x R (th x /ch 2 x. th R (ii lim x ± th x ±, (. 6..9 th z z e z + z 2, sh z th (z/2 th z, th z 2 th 2z th z. 6..0 th : R (, th : (, R ( y (, (i th (y 2 log +y y. (ii (th (y. y 2 (iii y < th (y y 2+ 2+. 6.. ( z C, r > 0 e r + 2r 3.3.5, 6..9 36 : (i 0 < z < r 0 < z < r sh z z + 2 th z ( 2 2 (2 2 B z 2 (2!. si z i sh (iz, ta z i th z 2 z + ( 2 2 B z 2 (2!. (ii ( (2 2 B z 2 (2!. (iii 0 < z < r/2 th (iz ta, si, ta 6.. 6.2 α C, N ( ( ( α α 0, α(α (α + /! 2.4.5. ( α 6.2. ( α C, x (, ( + x α 36 3.3.5 r 2π ( α x (. 75

f(x 2.5.6 g(x 5.. ( g (x α( + x α, ( + xg (x αg(x. (2 ( + xf (x αf(x. ( + xf (x 5..7 ( + x 2.4.5 ( α x ( α α x αf(x. {( α ( α } ( + + x + (, (2 f g g f. (, f g z C e z 0 g(x exp(α log( + x 0. f/g D ((, ( f f g g f g g 2 0. (, f/g c ( c (f/g(0 /. 665 α +etc.? 6.2. x (,, m N (+x m ( ( m+ x ( 6.2.2 (( + x ±/2 N\{0} 2 (double factorial (2!! 3 5 (2, (2!! 2 4 6 (2. (!! 0!! N ( def. /2 a ( (2 3!! ( 2 (2!! 2 2 2,, ( def. /2 b ( (2!! ( 2 (2!! 2 2, 0. (a, (b x (, 6.2. + x + ( a x, + x ( b x. 6.2.2 sh : R R sh : R R (i sh (y log(y + + y 2. (ii(sh (y / + y 2. (iii y (, sh (y ( b y 2+ 2+ ( b 6.2.2. 76

6.3 2.3.5 f f f f (f /(f f 6.3. ( I R I I I, f : I R, f C(I D m ( I (m, I f > 0 (a f I J f(i J f : J I (b J J J f C(J D m ( J J f f > 0, (f /(f f. (6.2 (c m, f C(I C m ( I f C(J C m ( J. : (a: ( 5.4.6 f I ( 2.3.5 f : J I (b: ( f D ( J (6.2 f y J f (y I. f (f (y > 0. z y, z y f (z f (y, f (z f (y. f (z f (y z y f (z f (y f(f (z f(f (y f (f (y. ( (2 f D m ( J m m ( m 2 f D m ( J f D m ( I ( 5.2.4 f f D m ( J. J f f > 0 ( 5.2.3 (f /(f f D m ( J. f D m ( J (c: (b 6.3. f y J f f (y > 0 (6.2 2 f f (y y (f f (f (6.2 2 77

6.4 ( θ y θ [ π 2, π 2 ] y arc legth Arcsi 6.4. ( si : [ π 2, π 2 ] [, ] ( 6.. Arcsi 2.3.5 Arcsi : [, ] [ π 2, π 2 ] (a y (, (Arcsi y / y 2. (b y [, ] Arcsi y b y 2+ 2 +, b (2!! (2!! ( 2 2 2. (a: Arcsi y [ π 2, π 2 ] cos(arcsi y 0. (si (Arcsi y cos(arcsi y si 2 (Arcsi y y 2 y (, 6.3. (Arcsi y (si (Arcsi y. y 2 (b: y (, 6.2.2 ( + y ( b y, ( ( f(y 5..7 (2 f D ((, f (y y (, (Arcsi y (a b y 2. y 2 ( 2 (2 b y f (y. 5.4.6 (, Arcsi f c (. 0 Arcsi 0 f(0 0. y ± y y y (, 4.3. y sh : R R 6.4. 6.2.2(ii,(iii 6.2.2(ii,(iii 6.4. 78

6.4. ( cos : [0, π] [, ] ( 6.. Arccos 2.3.5 Arccos : [, ] [0, π] y [, ] Arcsi y + Arccos y π 2 Arccos Arcsi 6.4.2 a C, T a (x cos(aarccos x (x [, ] (i ( x 2 T a (x xt a(x + a 2 T a (x 0. (ii N T T (cos θ cos θ, T ( x ( T (x T 37 6.4.2 ( ta : ( π 2, π 2 R ( 6..5 Arcta 2.3.5 Arcta : R ( π 2, π 2 lim Arcta x ±π, (. y ± 2 (a y R (Arcta y + y 2. (b y [, +] Arcta y (a: y R ( y 2+. 2 + (ta (Arcta y / cos 2 (Arcta y + ta 2 (Arcta y + y 2. 6.3. (Arcta y (ta (Arcta y + y 2. (b: y ± y (, f(y 5..7 ( f D ((, y (, f (y ( y 2. y (, (Arcta y (a + y 2 ( y 2 ( f (y. 5.4.6 (, Arcta f c ( c Arcta 0 f(0 0. th : R (, 6.4.2 6..0 (ii,(iii 6..0 (ii,(iii 6.4.2 6.4.3 (i x > 0 π 2 2Arcta x Arcta x2 2x. (ii ( ta(π/6 < x < ta(3π/6 π 4 4Arcta x + Arcta x4 +4x 3 6x 2 4x+ x 4 4x 3 6x 2 +4x+. (ii π 4 4Arcta 5 Arcta 239. Arcta 6.4.2 π 37 Pafuti i L vovich Chebyshev,82 94. 79

6.5 ( 6.5. z e z {z C ; Im z ( π, π} C\(, 0] Log (a z Log z C\(, 0] {z C ; Im z ( π, π}, (a2 z C\(, 0] e Log z z. (a3 z C, Im z ( π, π Log (e z z, (a4 z z e iθ C\{0}, θ ( π, π Log z log z + iθ. (b z Log z C\(, 0] (a0: 6..3 z e z {z C ; Im z [ π, π} C\{0} Im z π (, 0. (a (a3: (a0 (a4: (a3 (b: θ : C\(, 0] ( π, π 38 z x + iy (x, y R Arcta ( y x, x > 0, θ(z π/2 + Arcta x y, x 0, y > 0, π/2 Arcta, x 0, y < 0. ( 6.5., 6.5.2 ( e iθ(z z/ z, (2 θ C(C\(, 0]. (, (a4 z C\(, 0] x y Log z log z + iθ(z. (2 6..3 z e z {z C ; Im z [ π, π} C\{0} (, 0 (, 0] (, 0] Log 6.5. 6.5. ( 6.5.2 6.5. (2 6.5.3 α, z C, z < exp(αlog ( + z ( α z ( 38 θ(z z ( π, π (, (2 80

6.5.2 x [, ] Arcsi x i Log ( x 2 + ix. Arcsi x [ π/2, π/2] cos(arcsi x x 2. e iarcsi x cos(arcsi x + i si(arcsi x x 2 + ix C\(, 0]. Log 6.5.(a4 ( 6.5.4 x R Arcta x 2i Log +ix ix. 5.4.8 ( z 6.5.3 z C, z < Log ( + z x, y R, z x + iy, z <, (. 6.5. (a4 f(z Log ( + z, g(z ( z f(z 2 log(( + x2 + y 2 + iarcta 6.5.5 ( f x (z + z, f y (z z < g(z i + z. y + x. (2 g x (z }{{} 5.2.5 ( z + z }{{} ( f x (z. z < (3 g y (z }{{} i g x (z i }{{} + z f }{{} y (z. 5.2.5 (2 ( z < f(z g(z c ( z 0 c f(0 g(0 0 6.5.5 6.5.3 ( 8

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