A- Find all values of z such that sinh(z) = e'ő. B- Are |z], Img(z) analytic? Give reason.

Q: Determine the value of f(z) = e*cos y+ ie*sin y at z = In 4+ Ti/6

A:

Q: Let F = Vowhere o = e cos (Vy ). If C is the cardioid defined by ((2+ sin 0) sin 0, (2 + sin 0) cos…

A:

Q: Use the Fundamental Theorem of Calculus to find (2 + cos x) dx.

A:

Q: Determine the value of f(z)=e*cos y+ ie*sin y at z = In 4+ Ti/6.

A: For a complex number, z=x+iy , x is the real part and y is the imaginary part .

Q: Plot c(t) = (sin 2t, 2 cos t) for 0 ≤ t ≤ π. Express the length of the curve as a definite integral,…

A: Given, c(t) = (sin2t, 2cost) for 0 ≤ t ≤ π

Q: Prove that  d/dx[cosx]=-sinx

A: We will use the definition of the derivative  Here f(x)=cos(x) So,

Q: tan z, + tanz, = T-(tan z;) +(tan 72) tan(z, +2) ) Prove that: cos(z + w) = cosz cosw + sinz sin w

A:

Q: Prove the hyperbolic identity of sinhxcoshy-coshxsinhy=sinh(x-y)

A:

Q: S cose de 1-sine sece tane de sece - 1 The integral fo fse Si x dx x²+√√√x²−1−1 sino de 1+ cose…

A: we have to above integration

Q: to tal differential of eind the the り23 function ] 2=e + tan Z= Sin 2.

A:

Q: d Prove that 1 da (sin-1æ) = V1 – a? -

A:

Q: 2. F(x) = [ cosr' dt

A: Use fundamental theorem of calculus

Q: can you prove it, please ? cosh 2x=sinh^2x+1 = 2 cosh^2-1

A:

Q: Hiw/ If isinh Z= siniz, proveThat: sinhz= sinhxcosy +ising cosh x

A:

Q: #2. Verify Clairaut's Theorem for the given function f (s, t)= In (st)-s* sin (6t) +st.

A:

Q: Evaluate. d [arctan a – In ¤] dæ 1 1+#? 1 sec2 x sec² ¤ 1-x²

A: As we know that: The sum rule: f±g'=f'±g' The chain rule: fgx'=f'gx·g'x The Power law: ddxxn=nxn-1…

Q: Use the addition formulas for sinh x and cosh x to prove tanh u + tanh v tanh(u + v) = 1+ tanh u…

A:

Q: cos (In z) sin (In x) dx

A: To Evaluate the integral

Q: By Cauchy's integral formula show that, Se" esin e -2n cos(cose) de = 2n; "esine sin(cose) de = 0 0,

A: Cauchy's integral formula : Let U be an open subset of the complex plane C, and suppose the closed…

Q: find: de So 2-cose using sediment theorem

A: Solution :-

Q: 1 Proof (I): (sinhx)' = Vx? +1

A: For direct Proof:

Q: None Q2) prove that cosh(x + y) = coshx coshy + sinhx sinhy

A:

Q: Prove: L{cos at} = s2+ a?

A: We will prove the given statement.

Q: use a geoetry formula to find the exact value of the definite interval (-2x+4) dx on the interval…

A: The given function is (–2x+4) with the interval [0,3].

Q: 7. To evaluate f-cos X-1 dx, the integrand can be transformed to? 1-sin? x

A: Follow the procedure given below.

Q: Use the Fundamental Theorem to evaluate the definite integral exactly. 16 Enter the exact answer. 16…

A: Given query is to find the value of the integral.

Q: Select the definitions of sinhx, tanh a, and sech z. O e +e- e -e-z O e-e e +e-z O e -e e +e- O e +e…

A: The hyperbolic function of sine is expressed as-     sinh x=ex-e-x2 Thus, Option (C) is the correct…

Q: Evaluate si (z* - tan(z) cos(z) ) dx 2-1

A: To find the integral: ∫11x2-tanxcosxx3-1dx

Q: a. Show that If z = x+ iy where y 1 b. Evaluate: cosh?(-z) – sinh²(z)

A: Given that z =x+iy For a part , y<0 and we need to prove that | eiz | >1 For b part, we need…

Q: a. Show that If z=x+ iy where y 1 b. Evaluate: cosh (iz) + sin (2)

A:

Q: The integral S (A S- (в x dx x² + √√√x²-1- sino do 1 + cos sec²0 do 1+tano " cose de 1-sino sece…

A: Use trigonometric substitution

Q: Hico/ prove Thall Cosh2: - 2 tañ z Sinhz: 1+2 Tanhiz =

A: Since you have asked for a question with multiple sub-parts , we will solve the first three…

Q: Show that z- sinh z Rcs :* sinh z

A:

Q: Consider J sin'x cos²x dx. If we let "=cos x , the integrand can be transformed to. A 2u2 - - | (u²-…

A: The given integral,     ∫sin5x cos2x dx

Q: (a) Prove the Calculus identity sin- (a) + sin-(VT- 2²) = 5, z€ (0,1], where sin-, denotes the usual…

A: We’ll answer the first part of this question since due to complexity. Please submit the question…

Q: a z given that z С. Show that sin-1У ôxôy ôyôx

A:

Q: Find all solutions to cosh z = i sinh z.

A: To find all the solutions of the below equation. coshz=isinhz

Q: Prove the following hyperbolic trig identity. cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y).

A: cosh(x+y)=cosh(x)cosh(y)+sinh(x)sinh(y)

Q: 2. Prove that, V{(sec 0 - 1)/(sec 0 + 1)} cot 0. = cosec e

A:

Q: (a) Use the definitions above to show that d tanh x dr 1 cosh r

A:

Q: Prove that d/dz sin(z) = cos(z) using Cauchy-Riemann equations

A: Cauchy-Riemann Equation- If f(z)=u(x, y)+i v(x,  y) is analytic, then In particular, These…

Q: Prove that d/dx [cos x] = −sin x.

A:

Q: sinh (at)

A:

Q: a sin a + In v ) da + tan adv = 0 Cos a

A:

Q: b) Evaluate the integral: S (sin-1(x))²dx

A:

Q: Evaluate using Cauchy's Integral Theory: OP 3+ 2 cos (0) + sin (0)

A:

Q: Let = (sin x cos y, cos I sin y, In z).- Compute div i %3D

A: Given,          v→ =  sinxcosy , cosxsiny , lnz

Q: Use Gaussian Quadrature (n = 3) to approximate f(x)dx, where: f(x) = (2.54 - sin0.32x)2 cos0.32x ,…

A: