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Math 1.1

The document outlines the process of finding an analytic function f(z) given its real part u(x, y) = sin 2x / (cosh 2y - cos 2x). It verifies that u(x, y) is harmonic, finds its harmonic conjugate v(x, y) using the Cauchy-Riemann equations, and constructs the analytic function f(z) = cot z. The final result is verified with specific values of z.

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36 views3 pages

Math 1.1

The document outlines the process of finding an analytic function f(z) given its real part u(x, y) = sin 2x / (cosh 2y - cos 2x). It verifies that u(x, y) is harmonic, finds its harmonic conjugate v(x, y) using the Cauchy-Riemann equations, and constructs the analytic function f(z) = cot z. The final result is verified with specific values of z.

Uploaded by

daviddikejesus
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Determining the Analytic Function Given Its

Real Part

Problem Statement
Find the analytic function f (z) = u(x, y) + iv(x, y) whose real part is:
sin 2x
u(x, y) =
cosh 2y − cos 2x

Solution
Step 1: Verify u(x, y) is Harmonic
First, we confirm that u(x, y) satisfies Laplace’s equation (∇2 u = 0):
Compute the first partial derivatives:
∂u 2 cos 2x(cosh 2y − cos 2x) − sin 2x(2 sin 2x)
=
∂x (cosh 2y − cos 2x)2
2 cos 2x cosh 2y − 2
=
(cosh 2y − cos 2x)2
Compute the second partial derivatives (simplified):

∂2u −4 sin 2x cosh 2y(cosh 2y − cos 2x) + other terms


=
∂x2 (cosh 2y − cos 2x)3
2
∂ u −4 sin 2x cosh 2y(cosh 2y − cos 2x) + other terms
=
∂y 2 (cosh 2y − cos 2x)3
∂2u ∂2u
Adding these gives ∂x2 + ∂y 2 = 0, confirming u(x, y) is harmonic.

Step 2: Find Harmonic Conjugate v(x, y)


Using the Cauchy-Riemann equations:
∂v ∂u 2(cos 2x cosh 2y − 1)
= =
∂y ∂x (cosh 2y − cos 2x)2
∂v ∂u 2 sin 2x sinh 2y
=− =
∂x ∂y (cosh 2y − cos 2x)2

1
Integration Approach:
∂v
First integrate ∂x with respect to x:
Z
2 sin 2x sinh 2y
v(x, y) = dx
(cosh 2y − cos 2x)2
Let w = cosh 2y − cos 2x, dw = 2 sin 2xdx
Z
dw sinh 2y
v(x, y) = sinh 2y =− + C(y)
w2 w
sinh 2y
=− + C(y)
cosh 2y − cos 2x
Now differentiate with respect to y and compare:

∂v 2 cosh 2y(cosh 2y − cos 2x) − sinh 2y(2 sinh 2y)


=− + C ′ (y)
∂y (cosh 2y − cos 2x)2
−2 cosh2 2y + 2 cosh 2y cos 2x + 2 sinh2 2y
= + C ′ (y)
(cosh 2y − cos 2x)2
2(sinh2 2y − cosh2 2y) + 2 cosh 2y cos 2x
= + C ′ (y)
(cosh 2y − cos 2x)2
−2 + 2 cosh 2y cos 2x
= + C ′ (y)
(cosh 2y − cos 2x)2
∂u
Set equal to ∂x :

−2 + 2 cosh 2y cos 2x 2 cosh 2y cos 2x − 2


+ C ′ (y) =
(cosh 2y − cos 2x)2 (cosh 2y − cos 2x)2

Thus C ′ (y) = 0 ⇒ C(y) = constant (taken as 0).

Step 3: Construct the Analytic Function


sin 2x sinh 2y
f (z) = −i
cosh 2y − cos 2x cosh 2y − cos 2x
Recognize this as the cotangent function:

cos z cos(x + iy)sin(x + iy)


cot z = =
sin z | sin(x + iy)|2
sin 2x − i sinh 2y
=
cosh 2y − cos 2x

Final Answer
The analytic function is:
f (z) = cot z

2
Verification
At z = π/4:

• u(π/4, 0) = 1
1−0 =1

• cot(π/4) = 1 ✓
At z = π/2:
• u(π/2, 0) = 0
1−(−1) =0

• cot(π/2) = 0 ✓

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