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A Theorem Related to z*
? ?
22
11
uv
xy
vu
xy
z z* z z*
z x iy z* x iy x , y
i
x y z z*
u z,z*
u z u z*
x z x z* x
,
??
?
??
??
??
??
??
? ? ? ? ? ?
?
? ? ? ?
?
?
?
? ? ? ? ?
??
 C.R.
con
 C.R. conditions :
Note that 
 Make the above substitution for and and treat  and 
as independent variables :
?
? ?
v z,z*
v z v z*
y z y z* y
u u v v
i
z z* z z*
ii
?
? ? ? ?
??
? ? ? ? ?
?? ? ? ? ?
? ? ?
?
?
??
? ? ?
??
? ? ?
ds.
(1)
If f = f (z,z*) is analytic, then
0
f
z*
?
?
?
(The function cannot really vary with z*, and therefore cannot 
really be a function of z* except in a trivial way.)
Page 2


A Theorem Related to z*
? ?
22
11
uv
xy
vu
xy
z z* z z*
z x iy z* x iy x , y
i
x y z z*
u z,z*
u z u z*
x z x z* x
,
??
?
??
??
??
??
??
? ? ? ? ? ?
?
? ? ? ?
?
?
?
? ? ? ? ?
??
 C.R.
con
 C.R. conditions :
Note that 
 Make the above substitution for and and treat  and 
as independent variables :
?
? ?
v z,z*
v z v z*
y z y z* y
u u v v
i
z z* z z*
ii
?
? ? ? ?
??
? ? ? ? ?
?? ? ? ? ?
? ? ?
?
?
??
? ? ?
??
? ? ?
ds.
(1)
If f = f (z,z*) is analytic, then
0
f
z*
?
?
?
(The function cannot really vary with z*, and therefore cannot 
really be a function of z* except in a trivial way.)
? ? ? ?
11
v z,z* u z,z*
v z v z* u z u z*
x z x z* x y z y z* y
v v u u
i
z z* z z*
f u v v
ii
z* z* z* z
i
i
??
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
?? ? ? ? ?
? ? ? ?
??
? ? ? ?
??
? ? ? ?
? ? ? ?
? ? ? ?
?
??
??
?
 C.R.
conds.
Similarly,
(2)
Next, consider 
?
vu
z* z
??
??
??
??
??
??
v
i
z
?
?
?
from (2)
u
z
?
?
?
0
u
z*
u v f f
* f
i
z* z* z* z*
z
??
?
?
??
?
??
?? ? ? ? ?
? ? ? ? ? ?
??
? ? ?
??
?
?
?
from (1)
is independent of 
A Theorem Related to z* (cont.)
Page 3


A Theorem Related to z*
? ?
22
11
uv
xy
vu
xy
z z* z z*
z x iy z* x iy x , y
i
x y z z*
u z,z*
u z u z*
x z x z* x
,
??
?
??
??
??
??
??
? ? ? ? ? ?
?
? ? ? ?
?
?
?
? ? ? ? ?
??
 C.R.
con
 C.R. conditions :
Note that 
 Make the above substitution for and and treat  and 
as independent variables :
?
? ?
v z,z*
v z v z*
y z y z* y
u u v v
i
z z* z z*
ii
?
? ? ? ?
??
? ? ? ? ?
?? ? ? ? ?
? ? ?
?
?
??
? ? ?
??
? ? ?
ds.
(1)
If f = f (z,z*) is analytic, then
0
f
z*
?
?
?
(The function cannot really vary with z*, and therefore cannot 
really be a function of z* except in a trivial way.)
? ? ? ?
11
v z,z* u z,z*
v z v z* u z u z*
x z x z* x y z y z* y
v v u u
i
z z* z z*
f u v v
ii
z* z* z* z
i
i
??
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
?? ? ? ? ?
? ? ? ?
??
? ? ? ?
??
? ? ? ?
? ? ? ?
? ? ? ?
?
??
??
?
 C.R.
conds.
Similarly,
(2)
Next, consider 
?
vu
z* z
??
??
??
??
??
??
v
i
z
?
?
?
from (2)
u
z
?
?
?
0
u
z*
u v f f
* f
i
z* z* z* z*
z
??
?
?
??
?
??
?? ? ? ? ?
? ? ? ? ? ?
??
? ? ?
??
?
?
?
from (1)
is independent of 
A Theorem Related to z* (cont.)
? ?
? ? ? ?
? ?
? ?
2
* 1 0 *
*
sin * cos * 0 2 1 / 2
*
* 0 0
*
*
*
f
f z z z
z
f
f z z z z n
z
f
f z z zz z z
z
z
fz
z
?
?
? ? ?
?
?
? ? ? ? ?
?
?
? ? ? ? ?
?
?
unless
unless
is analytic nowhere, since (not independent of )
 is not analytic, since ( )
is not analytic, since ( )
is  anal
Examples:
? ?
10
**
f
zz
??
??
??
ytic everywhere, since 
A Theorem Related to z* (cont.)
Page 4


A Theorem Related to z*
? ?
22
11
uv
xy
vu
xy
z z* z z*
z x iy z* x iy x , y
i
x y z z*
u z,z*
u z u z*
x z x z* x
,
??
?
??
??
??
??
??
? ? ? ? ? ?
?
? ? ? ?
?
?
?
? ? ? ? ?
??
 C.R.
con
 C.R. conditions :
Note that 
 Make the above substitution for and and treat  and 
as independent variables :
?
? ?
v z,z*
v z v z*
y z y z* y
u u v v
i
z z* z z*
ii
?
? ? ? ?
??
? ? ? ? ?
?? ? ? ? ?
? ? ?
?
?
??
? ? ?
??
? ? ?
ds.
(1)
If f = f (z,z*) is analytic, then
0
f
z*
?
?
?
(The function cannot really vary with z*, and therefore cannot 
really be a function of z* except in a trivial way.)
? ? ? ?
11
v z,z* u z,z*
v z v z* u z u z*
x z x z* x y z y z* y
v v u u
i
z z* z z*
f u v v
ii
z* z* z* z
i
i
??
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
?? ? ? ? ?
? ? ? ?
??
? ? ? ?
??
? ? ? ?
? ? ? ?
? ? ? ?
?
??
??
?
 C.R.
conds.
Similarly,
(2)
Next, consider 
?
vu
z* z
??
??
??
??
??
??
v
i
z
?
?
?
from (2)
u
z
?
?
?
0
u
z*
u v f f
* f
i
z* z* z* z*
z
??
?
?
??
?
??
?? ? ? ? ?
? ? ? ? ? ?
??
? ? ?
??
?
?
?
from (1)
is independent of 
A Theorem Related to z* (cont.)
? ?
? ? ? ?
? ?
? ?
2
* 1 0 *
*
sin * cos * 0 2 1 / 2
*
* 0 0
*
*
*
f
f z z z
z
f
f z z z z n
z
f
f z z zz z z
z
z
fz
z
?
?
? ? ?
?
?
? ? ? ? ?
?
?
? ? ? ? ?
?
?
unless
unless
is analytic nowhere, since (not independent of )
 is not analytic, since ( )
is not analytic, since ( )
is  anal
Examples:
? ?
10
**
f
zz
??
??
??
ytic everywhere, since 
A Theorem Related to z* (cont.)
Entire Functions
12
22
2 3 4 5
1
sin cos sinh cosh
11
, tan
1
/
n
z
, z, z , z , z ,z , ,z ,
e , z, z, z, z
, z ,
zz ?
   Typical functionsthat are entire
 (analytic everywhere in the finite complex plane):
   Typical functions analytic everywhere : almost
 cot tanh coth z, z, z, z
A function that is analytic everywhere in the finite* 
complex plane is called “entire”.
* A function is said to be analytic everywhere in the finite complex plane if 
it is analytic everywhere except possibly at infinity.
1 w / z ? Let Analytic at infinity: Is the function analytic at w = 0?
Page 5


A Theorem Related to z*
? ?
22
11
uv
xy
vu
xy
z z* z z*
z x iy z* x iy x , y
i
x y z z*
u z,z*
u z u z*
x z x z* x
,
??
?
??
??
??
??
??
? ? ? ? ? ?
?
? ? ? ?
?
?
?
? ? ? ? ?
??
 C.R.
con
 C.R. conditions :
Note that 
 Make the above substitution for and and treat  and 
as independent variables :
?
? ?
v z,z*
v z v z*
y z y z* y
u u v v
i
z z* z z*
ii
?
? ? ? ?
??
? ? ? ? ?
?? ? ? ? ?
? ? ?
?
?
??
? ? ?
??
? ? ?
ds.
(1)
If f = f (z,z*) is analytic, then
0
f
z*
?
?
?
(The function cannot really vary with z*, and therefore cannot 
really be a function of z* except in a trivial way.)
? ? ? ?
11
v z,z* u z,z*
v z v z* u z u z*
x z x z* x y z y z* y
v v u u
i
z z* z z*
f u v v
ii
z* z* z* z
i
i
??
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
?? ? ? ? ?
? ? ? ?
??
? ? ? ?
??
? ? ? ?
? ? ? ?
? ? ? ?
?
??
??
?
 C.R.
conds.
Similarly,
(2)
Next, consider 
?
vu
z* z
??
??
??
??
??
??
v
i
z
?
?
?
from (2)
u
z
?
?
?
0
u
z*
u v f f
* f
i
z* z* z* z*
z
??
?
?
??
?
??
?? ? ? ? ?
? ? ? ? ? ?
??
? ? ?
??
?
?
?
from (1)
is independent of 
A Theorem Related to z* (cont.)
? ?
? ? ? ?
? ?
? ?
2
* 1 0 *
*
sin * cos * 0 2 1 / 2
*
* 0 0
*
*
*
f
f z z z
z
f
f z z z z n
z
f
f z z zz z z
z
z
fz
z
?
?
? ? ?
?
?
? ? ? ? ?
?
?
? ? ? ? ?
?
?
unless
unless
is analytic nowhere, since (not independent of )
 is not analytic, since ( )
is not analytic, since ( )
is  anal
Examples:
? ?
10
**
f
zz
??
??
??
ytic everywhere, since 
A Theorem Related to z* (cont.)
Entire Functions
12
22
2 3 4 5
1
sin cos sinh cosh
11
, tan
1
/
n
z
, z, z , z , z ,z , ,z ,
e , z, z, z, z
, z ,
zz ?
   Typical functionsthat are entire
 (analytic everywhere in the finite complex plane):
   Typical functions analytic everywhere : almost
 cot tanh coth z, z, z, z
A function that is analytic everywhere in the finite* 
complex plane is called “entire”.
* A function is said to be analytic everywhere in the finite complex plane if 
it is analytic everywhere except possibly at infinity.
1 w / z ? Let Analytic at infinity: Is the function analytic at w = 0?
Combinations of Analytic Functions
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ?
f z ,g z ,h z
a f z b g z c h z
f z ,g z
f g z
? ?
?
?
   of analytic functions are analytic :
are analytic
is analytic
 of analytic functions are analytic :
are analytic
is analy
If
If
Finite linear combinations
Composite combinations
tic 
Combinations of functions:

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FAQs on PPT: Analytic Functions - Engineering Mathematics - Civil Engineering (CE)

1. What are analytic functions?

Ans. Analytic functions are mathematical functions that can be represented by convergent power series expansions. They are defined in complex analysis and have the property of being differentiable at every point in their domain.

2. What is the significance of analytic functions?

Ans. Analytic functions are important in various fields of mathematics and physics. They play a crucial role in complex analysis, as they satisfy the Cauchy-Riemann equations and have many useful properties. They are also used in the study of harmonic functions, potential theory, and the theory of functions of a complex variable.

3. How can analytic functions be classified?

Ans. Analytic functions can be classified into several types based on their properties. Some common classifications include entire functions (functions that are analytic in the entire complex plane), meromorphic functions (functions that are analytic except for isolated singularities), and rational functions (functions that can be expressed as the ratio of two polynomials).

4. What is the relationship between analytic functions and holomorphic functions?

Ans. Analytic functions and holomorphic functions are closely related. In complex analysis, a function is said to be analytic if it is differentiable at every point in its domain. Holomorphic functions, on the other hand, are functions that are complex differentiable at every point in an open subset of the complex plane. It can be shown that every analytic function is holomorphic, but not every holomorphic function is analytic.

5. What are some examples of analytic functions?

Ans. There are many examples of analytic functions, including polynomial functions, exponential functions, trigonometric functions, and logarithmic functions. These functions have well-defined power series expansions and are differentiable at every point in their domains.

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