(i)(1/2)-(-i)(2) 2i Section 12 The Exponential and Trigonometric Functions 69 Notice from both these examples that sines and cosines of complex numbers may be greater than 1. As we shall see (Section 15), although | sin z| < 1 and cos r< 1 for real r, when z is a complex number, sin z and cos z can have any value we like. Using the definitions (11.4) of sin z and cos z, you can show that the familiar trigonometric identities and calculus formulas hold when 0 is replaced by z Example 3. Prove that sin2 z + cos2 z = 1. e2iz-2e-2iz e-eiz2 sin z= 2i e2iz +2+e-2iz cos z sin zcos z = Example 4. Using the definitions (11.4), verify that (d/dz) sinz = cos z. eize ie) ei sin z= PROBLEMS, SECTION 11 Define sin z and cos z by their power series. Write the power series for e. By comparing these series obtain the definition (11.4) of sinz and cos z. Solve the equations e = cos0isin 0, e-i cos 0-isin 0, for cos 0 and sin 0 and so obtain equations (11.3). 1. 2. Find each of the following in rectangular form r+iy and check your results by computer. Remember to save time by doing as much as you can in your head (in/4)+(Ia 2)/2 e-(in/4)+ln 3 3 la 2-i 4 3 6. cos(in 5) 9. sin(-iIn 3) cos(-2i In 3) 7. tan(in 2) 10. sin(iIn i In the following integrals express the sines and cosines in exponential form and then integrate to show that: Cos2 3r drT cOs 2z cos 3r dr=0 sin 4r dr=T 13. sin 2r sin 3r dr= 0 14. 15. sin 2r cos 3 dz = 0 sin 3r cos 4r dr= 0 Evaluate fea+ib)adz and take real and imaginary parts to show that: ,ar Cos br dr= a cos br + b sin br) a2b2 a(a sin br- b cos br) a2 +b2
(i)(1/2)-(-i)(2) 2i Section 12 The Exponential and Trigonometric Functions 69 Notice from both these examples that sines and cosines of complex numbers may be greater than 1. As we shall see (Section 15), although | sin z| < 1 and cos r< 1 for real r, when z is a complex number, sin z and cos z can have any value we like. Using the definitions (11.4) of sin z and cos z, you can show that the familiar trigonometric identities and calculus formulas hold when 0 is replaced by z Example 3. Prove that sin2 z + cos2 z = 1. e2iz-2e-2iz e-eiz2 sin z= 2i e2iz +2+e-2iz cos z sin zcos z = Example 4. Using the definitions (11.4), verify that (d/dz) sinz = cos z. eize ie) ei sin z= PROBLEMS, SECTION 11 Define sin z and cos z by their power series. Write the power series for e. By comparing these series obtain the definition (11.4) of sinz and cos z. Solve the equations e = cos0isin 0, e-i cos 0-isin 0, for cos 0 and sin 0 and so obtain equations (11.3). 1. 2. Find each of the following in rectangular form r+iy and check your results by computer. Remember to save time by doing as much as you can in your head (in/4)+(Ia 2)/2 e-(in/4)+ln 3 3 la 2-i 4 3 6. cos(in 5) 9. sin(-iIn 3) cos(-2i In 3) 7. tan(in 2) 10. sin(iIn i In the following integrals express the sines and cosines in exponential form and then integrate to show that: Cos2 3r drT cOs 2z cos 3r dr=0 sin 4r dr=T 13. sin 2r sin 3r dr= 0 14. 15. sin 2r cos 3 dz = 0 sin 3r cos 4r dr= 0 Evaluate fea+ib)adz and take real and imaginary parts to show that: ,ar Cos br dr= a cos br + b sin br) a2b2 a(a sin br- b cos br) a2 +b2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
Problem, Section 11 number 5,8 and 9. Pliss answer the question
Transcribed Image Text:(i)(1/2)-(-i)(2)
2i
Section 12
The Exponential and Trigonometric Functions
69
Notice from both these examples that sines and cosines of complex numbers may
be greater than 1. As we shall see (Section 15), although | sin z| < 1 and cos r< 1
for real r, when z is a complex number, sin z and cos z can have any value we like.
Using the definitions (11.4) of sin z and cos z, you can show that the familiar
trigonometric identities and calculus formulas hold when 0 is replaced by z
Example 3. Prove that sin2 z + cos2 z = 1.
e2iz-2e-2iz
e-eiz2
sin z=
2i
e2iz +2+e-2iz
cos z
sin zcos z =
Example 4. Using the definitions (11.4), verify that (d/dz) sinz = cos z.
eize
ie) ei
sin z=
PROBLEMS, SECTION 11
Define sin z and cos z by their power series. Write the power series for e. By
comparing these series obtain the definition (11.4) of sinz and cos z.
Solve the equations e = cos0isin 0, e-i cos 0-isin 0, for cos 0 and sin 0 and
so obtain equations (11.3).
1.
2.
Find each of the following in rectangular form r+iy and check your results by computer.
Remember to save time by doing as much as you can in your head
(in/4)+(Ia 2)/2
e-(in/4)+ln 3
3 la 2-i
4
3
6. cos(in 5)
9. sin(-iIn 3)
cos(-2i In 3)
7. tan(in 2)
10. sin(iIn i
In the following integrals express the sines and cosines in exponential form and then
integrate to show that:
Cos2 3r drT
cOs 2z cos 3r dr=0
sin 4r dr=T
13.
sin 2r sin 3r dr= 0
14.
15.
sin 2r cos 3 dz = 0
sin 3r cos 4r dr= 0
Evaluate fea+ib)adz and take real and imaginary parts to show that:
,ar Cos br dr= a cos br + b sin br)
a2b2
a(a sin br- b cos br)
a2 +b2
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