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The PDE 2∂
2
z
∂x
2
+2∂
2
z
∂y
2
+4∂
2
z
∂x∂y
=2x
2∂2z∂x2+2∂2z∂y2+4∂2z∂x∂y=2x,
is
ind an analytic continuation 𝑓𝑡 ,𝐷𝑡 : 0 ≤ 𝑡 ≤ 1 of
𝑓0
,𝐷0
along 𝛾 and show that 𝑓1
1 = 𝑓0
1
Where 𝐷0 = 𝐵 1 , 1 𝑎𝑛𝑑 𝑓0
is restriction of the principal
branch of 𝑧 to 𝐷0
. 𝛾 𝑡 = 𝑒
2𝜋𝑖𝑡 𝑎𝑛𝑑 𝜎 𝑡 = 𝑒
4𝜋𝑖𝑡
Find z1/n; for n=3, z = 1-i in C (, the Argand Plane).
Show that f is analytic on D iff integral -π to π 𝑓(𝑒^
𝑖𝑡
)𝑒^
𝑖𝑡𝑑𝑡 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥ 1
If f(z) is an entire function such that f(z)=f(-z) then there exists an entire function g(z) such that f(z)=g(z^2)
Use Cauchy’s integral formula to evaluate sin(z) /((2z + 1)*3e*z) dz
1 Prove that a necessary condition that
w = f(z) = u(x, y) + iv(x, y)
be analytic in a region R is that the Cauchy-Riemann equations
∂u
∂y =
∂v
∂y ,
∂u
∂y = −
∂v
∂x
are satisfied in a region R where it is supposed that these partial derivatives are
continuous in R
Express the following in rectangular and polar form, if
Z1 = 3+ 4i
Z2= 2+3i
1. Z1*Z2
2. Z1-Z2
3. Z1/Z2
4. |Z1|
5. |Z2|
(2)if Z1=50<30°and Z2=30<60°find in rectangular
form the following
1. |Z1|
2. |Z2|
3. |Z1|-|Z2|
4. Z1*Z2
5. |Z2-Z1|
6. |Z2|/|Z1|
Show that 𝑓(𝑧) = 𝑥 2 + 𝑦 2 where 𝑧 = 𝑥 + 𝑖𝑦 is not analytic anywhere using Cauchy Riemann equations.
Develop 1/(1+z^2) in powers of z-a, a being a real number.Find the general coefficient and for a=1 reduce to simplest form.
