P
Z 1
À1
e
iz
þ e
Àiz
z À b
dz ¼ À2π sin b:
Thus, the final answer is
I ¼
1
2 a À b
ð
Þ
Z 1
À1
e
iz
þ e
Àiz
z À a
À
e
iz
þ e
Àiz
z À b
dz ¼
2π sin b À sin a
ð
Þ
2 a À b
ð
Þ
¼
π sin b À sin a
ð
Þ
a À b
:
ð6:224Þ
Example 6.11 Evaluate the following real definite integral:
I ¼
Z 1
À1
cos x
1 þ x 2 dx:
ð6:225Þ
The calculation is left for readers. Hints: (i) Use cosx ¼ (e
ix + e
Àix
)/2 and Jordan’s
lemma. (ii) Use the contour as shown in Fig. 6.18 for integration. Besides the
examples listed above, a variety of integral calculations can be seen in literature
[9–12].
6.9 Multivalued Functions and Riemann Surfaces
The last topics of the theory of analytic functions are related to multivalued functions. Riemann surfaces play a central role in developing the theory of the
multivalued functions.
6.9.1 Brief Outline
We mention a brief outline for notions of multivalued functions and Riemann
surfaces. These notions are indispensable for investigating properties of irrational
functions and logarithmic functions and performing related calculations, especially
integrations with respect to those functions.
Definition 6.12 Let n be a positive integer and let z be an arbitrary complex number.
Suppose we have
z ¼ w
n
:
ð6:226Þ
Then, w is said to be a n-th root of z and we denote
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6 Theory of Analytic Functions
Z 1
À1
e
iz
þ e
Àiz
z À b
dz ¼ À2π sin b:
Thus, the final answer is
I ¼
1
2 a À b
ð
Þ
Z 1
À1
e
iz
þ e
Àiz
z À a
À
e
iz
þ e
Àiz
z À b
dz ¼
2π sin b À sin a
ð
Þ
2 a À b
ð
Þ
¼
π sin b À sin a
ð
Þ
a À b
:
ð6:224Þ
Example 6.11 Evaluate the following real definite integral:
I ¼
Z 1
À1
cos x
1 þ x 2 dx:
ð6:225Þ
The calculation is left for readers. Hints: (i) Use cosx ¼ (e
ix + e
Àix
)/2 and Jordan’s
lemma. (ii) Use the contour as shown in Fig. 6.18 for integration. Besides the
examples listed above, a variety of integral calculations can be seen in literature
[9–12].
6.9 Multivalued Functions and Riemann Surfaces
The last topics of the theory of analytic functions are related to multivalued functions. Riemann surfaces play a central role in developing the theory of the
multivalued functions.
6.9.1 Brief Outline
We mention a brief outline for notions of multivalued functions and Riemann
surfaces. These notions are indispensable for investigating properties of irrational
functions and logarithmic functions and performing related calculations, especially
integrations with respect to those functions.
Definition 6.12 Let n be a positive integer and let z be an arbitrary complex number.
Suppose we have
z ¼ w
n
:
ð6:226Þ
Then, w is said to be a n-th root of z and we denote
248
6 Theory of Analytic Functions