Verify Cauchy Riemann equations are satisfied function \sin h 4z

Cabiolab

Cabiolab

Answered question

2021-05-18

Verify Cauchy Riemann equations are satisfied function sinh 4z

Answer & Explanation

estenutC

estenutC

Skilled2021-05-20Added 81 answers

Step 1
Given that, f(z)=sinh(4z)
To verify, Cauchy Riemann equations are satisfied for the function f(z)=sinh(4z)
We know that z=x+iy
f(z)=sinh(4(x+iy))
=sinh(4x+4iy)
=sinh(4x)cosh(i4y)+cosh(4x)sinh(4y)
=sinh(4x)cos(4y)+icosh(4x)sin(4y)
Here real part u=sinh(4x)cos(4y), imaginary part v=cosh(4x)sin(4y)
Step 2
Check whether ux=vy and uy=vx
u=sinh(4x)cos(4y)
uy=4sinh(4x)sin(4y)
ux=4cosh(4x)cos(4y)
v=cosh(4x)sin(4y)
vx=4sinh(4x)sin(4y)
vy=4cosh(4x)cos(4y)
Therefore, ux=vy and uy=vx
Hence, The function f(z)=sinh(4z) satisfies the Cauchy Riemann equations.
Jeffrey Jordon

Jeffrey Jordon

Expert2021-11-20Added 2605 answers

Answer is given below (on video)

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