Combined Elementary transformation : LINEAR TRANSFORMATION : 𝑤 = 𝛼𝑧+𝛽, Example:- 𝑤 = (1+𝑖)𝑧+(2-𝑖) .

Complex Analysis Theorem from Conformal transformation
(Combined Elementary transformation )

𝑳𝑰𝑵𝑬𝑨𝑹 𝑻𝑹𝑨𝑵𝑺𝑭𝑶𝑹𝑴𝑨𝑻𝑰𝑶𝑵 :-
The transformation of the form
𝑤 = 𝛼𝑧+𝛽
Where 𝛼 and 𝛽 both are complex constant then this transformation is known as a linear transformation .
This transformation is the resultant of magnification, rotation and translation.

𝑬𝑿𝑨𝑴𝑷𝑳𝑬 :-

Consider a linear transformation
𝑤 = (1+𝑖)𝑧+(2-𝑖) and determine the region in 𝑹' in 𝑤 plane into which the corresponding rectangular region 𝑹 bounded by 𝑥=0, 𝑦=0, 𝑥=1 and 𝑦=2 and in 𝑧 plane.

𝐒𝐎𝐋𝐔𝐓𝐈𝐎𝐍 :-
Given that
𝑤 = (1+𝑖)𝑧+(2-𝑖)
𝑢+𝑖𝑣=(1+𝑖)(𝑥+𝑖𝑦)+(2-𝑖)
𝑢+𝑖𝑣=(𝑥+𝑖𝑥+𝑖𝑦-𝑦+2-𝑖)
𝑢+𝑖𝑣=(𝑥-𝑦+2)+𝑖(𝑥+𝑦-1)
𝑢=(𝑥-𝑦+2) and 𝑣=(𝑥+𝑦-1)
∵ the rectangular region 𝑹 bounded by the lines 𝑥=0, 𝑦=0, 𝑥=2 and 𝑦=3 and in 𝑧 plane.
at 𝑥=0 then 𝑢=-𝑦+2 and 𝑣= 𝑦-1 ⇒ 𝑢+𝑣=1
∴ 𝑢=0, 𝑣=1 ⇒ (0,1) and 𝑢=1, 𝑣=0 ⇒ (1,0)
at 𝑦=0 then 𝑢=𝑥+2 and 𝑣=𝑥-1 ⇒ 𝑢-𝑣=3
∴ 𝑢=0, 𝑣=-3 ⇒ (0,-3) and 𝑢=3, 𝑣=0 ⇒ (3,0)
at 𝑥=1 then 𝑢=-𝑦+3 and 𝑣=𝑦 ⇒ 𝑢+𝑣=3
∴ 𝑢=0, 𝑣=3 ⇒ (0,3) and 𝑢=3, 𝑣=0 ⇒ (3,0)
at y=2 then 𝑢=𝑥 and 𝑣=𝑥+1 ⇒ 𝑢-𝑣=-1
∴ 𝑢=0, 𝑣=1 ⇒ (0,1) and 𝑢=-1, 𝑣=0 ⇒ (-1,0)
Hence
the required image is a rectangle region 𝑹' bounded by the lines 𝑢+𝑣=1, 𝑢-𝑣=3 , 𝑢+𝑣=3 and 𝑢-𝑣=-1 in 𝑤 plane corresponding to rectangular region 𝑹 bounded by 𝑥=0, 𝑦=0, 𝑥=1 and 𝑦=2 and in 𝑧 plane.
here ∵ 𝛼=(1+𝑖)=√2.𝑒ⁱ⁽ ⷫᐟ⁴⁾ the figure in 𝑤 plane is √2 times, rotate (𝝅/4) and 𝛽=(2-𝑖) origin shift (2,-1) and of figure of 𝑧 plane .
See figure in video.
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Solved...!!!
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#complexanalysis #csirnet #conformal
#isogonal #transformation #mapping #lineartransformation
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