- Suggested problems for Section 6.6.1: 1-13. (some solutions)
- Suggested problems for Section 6.1: 15-20, 21-30, 35.
- Suggested problems for Section 6.2: 1, 2, 4, 6, 7, 8, 11, 12, 13, 14, 16, 18, 20, 25, 26, 27, 29, 30.
- Suggested problems for Section 5.4: 4, 10, 12, 13, 16, 17-19, 22, 23.
- Suggested problems for Section 5.5: 1-24, in particular 21, 22.
- Suggested problems for Section 5.2: 3, 5, 7, 8, 11, 12, 15, 21, 24, 29, 30.
- Suggested problems for Section 5.3: 2-10, 11-21, 24, 29, 30.
- Suggested problems for Section 4.3: 1, 3, 4, 5, 9-16, 17, 19, 21, 22, 25, 26, 28, 29, 33, 40, 45, 50.
- Suggested problems for Section 4.4: 1, 2, 3, 6, 7, 8, 19.
- Suggested problems for Section 4.1: 1, 3, 4, 5, 7, 8, 9, 11-14, 15-20, 21, 22, 24, 27, 28, 32, 33-36, 38, 40, 41-46, 49, 51, 52.
- Suggested problems for Section 4.2:1, 4, 5, 7-11, 20
- Updated Mathematica file ColoredGraph.nb.
- Suggested problems for Section 3.1: 2-5, 15-19, 21, 22, 23, 25, 27-30, 33, 37.
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Suggested problems for Section 3.2: 1, 2, 3, 6-9, 9, 11, 13, 15, 17, 18, 23, 24, 25-28, 30, 31, 33, 34, 35. In problems 9, 11, 13, 15 you can find a simple formula for
in terms of
only. Then you can apply rules of differentiation to get the derivative. Compare the answer obtained in this way to the one obtained following the hint given in the book.
- Suggested problems for Section 3.3: 1, 3, 7, 11-14, 18.
-
The complex exponential function is defined by the following formula
\[ \exp(z) = e^z = e^{x+i y} = e^x \cos(y) + i\, e^x \sin(y) \] Here are two views of the complex exponential function. The surface in the pictures is the graph of the function $u(x,y) = e^x \cos y.$ The coloring is determined by the number $e^x \sin y$; the negative numbers are colored in violet (this is the color which is a half way between blue and magenta, click here to learn more); the positive numbers are colored in chartreuse (this is the color which is a half way between yellow and green, click here to learn more); numbers with larger absolute value are lighter, the numbers with smaller absolute value are darker; zero being black.
- I keep updating the Mathematica file ColoredGraph.nb.
- Suggested problems for Section 2.6: 7, 13, 15, 17, 19, 31, 37, 38, 40, 41-44, 49, 50, 51.
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Complex root functions, such as the complex square root and the complex cube root, can be viewed as multivalued functions: for the square root to each nonzero complex number there correspond two values of the (multivalued) square root function, for the cube root to each nonzero complex number there correspond three values of the (multivalued) cube root function. The pictures below represent graphs of these functions in the four dimensional space. Two horizontal axes are real and imaginary part of the independent variable z. The vertical axes is the real part of the dependent variable w, while the imaginary part of w is represented by color, as in the complex plane as explained earlier. Place the cursor over the image to see the transformation from the graph of $w =z,$ through fractional powers to $w =\sqrt{z}$ and then further to $w =\sqrt[3]{z}.$
The multivalued square root function
Place the cursor over the image to start the animation.
The multivalued cube root function
Place the cursor over the image to start the animation.
- I have updated the Mathematica file ColoredGraph.nb. Now it is organized into sections. It includes the code for the above pictures. As before, right-click on the underlined name of the file; in the pop-up menu that appears your browser will offer you to save the file in your directory. Make sure that the file is saved with exactly this name: ColoredGraph.nb. After saving the file you can open it from Mathematica. Start Mathematica, then click on the menu item File, choose Open, and then find ColoredGraph.nb. When you open this file place the cursor in the first cell and press Shift+Enter. Then continue with other cells. You can get all the images which are presented above and many more.
- Suggested problems for Section 2.5: 5-10, 11-14, 16, 18, 20, 21, 22, 23, 25-30.
- Examples below show transformation of rectangular (standard) grids under power functions.
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Here is a transformation of the rectangle
\[
\frac{1}{2} \leq x \leq \frac{3}{2}, -\frac{1}{2} \leq y \leq \frac{3}{2}.
\]
Place the cursor over the image to see the transformation. The animation uses the following 25 rational powers: $z^{k/12}, k \in \{12,13,\ldots,36\}.$
Place the cursor over the image to start the animation.
-
Here is a transformation of a slightly changed rectangle:
\[
\frac{1}{2} \leq x \leq \frac{3}{2}, -\frac{3}{2} \leq y \leq \frac{3}{2}.
\]
Place the cursor over the image to start the animation.
-
Here is a transformation of the square:
\[
-\frac{3}{2} \leq x \leq \frac{3}{2}, -\frac{3}{2} \leq y \leq \frac{3}{2}.
\]
The images show the power that is used to generate a particular transformation.
Place the cursor over the image to start the animation.
- Here is the Mathematica file that I used to create pictures above. It is called ComplexGrids.nb. Right-click on the underlined word "here"; in the pop-up menu that appears your browser will offer you to save the file in your directory. Make sure that the file is saved with exactly this name: ComplexGrids.nb. After saving the file you can open it from Mathematica. Start Mathematica, then click on the menu item File, choose Open, and then find ComplexGrids.nb. When you open this file place the cursor in the first cell and press Shift+Enter. Then continue with other cells. You can get all the images which are presented above and many more.
- Suggested problems for Section 2.4: 3, 4, 8, 9, 10, 15, 17, 18, 22, 23, 24, 25-30, 31-36, 39, 40, 47, 48, 49.
- One way to visualize complex functions is to use colors, a lot of colors. The basic idea here is to associate a color with each complex number. I will use a variation of the coloring scheme defined by Jan Homann who provided a colored graph of the Riemann Zeta function to Wikipedia. I explain this coloring scheme in the next two items.
-
This coloring is based on the RGB color wheel. Here are three color wheels with different resolutions.
The argument of a complex number $z$ is encoded by the hue of a color.
Red stands for positive reals (that is, $\operatorname{Arg}(z) =0,$
and then counterclockwise through
yellow $\operatorname{Arg}(z) = \pi/3$), green ($\operatorname{Arg}(z) = 2\pi/3$),
cyan (negative reals, $\operatorname{Arg}(z) =\pi$),
blue ($\operatorname{Arg}(z) = -2\pi/3$) and magenta ($\operatorname{Arg}(z) = -\pi/3$). -
The modulus of a complex number is expressed by darkness or lightness of color. The numbers on the unit circle are colored by the colors from the color wheel. The numbers inside the unit circle are darker, the origin being black. The number outside the unit circle are lighter, the "infinity" being white. The following three pictures illustrate this in a somewhat exaggerated way.
-
Here is the colored complex plane. Place the cursor over the image to see a magnification.
-
Here is a colored graph of the function $\operatorname{Re}(z).$ Place the cursor over the image to see a magnification.
-
Here is a colored graph of the function $\operatorname{Im}(z).$ Place the cursor over the image to see a magnification.
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Here is a colored graph of the function $\operatorname{Arg}(z).$ No magnification is needed here.
-
Here is a colored graph of the function $w=|z|.$ Place the cursor over the image to see a magnification.
-
Here is a colored graph of the function $w=z^2.$ Place the cursor over the image to see a magnification.
-
Here is a colored graph of the function $w=z^3.$ Place the cursor over the image to see a magnification.
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And one more colored graph for today; here is a colored graph of the function $w=z^3-1.$ Place the cursor over the image to see a magnification.
- Finally, here is the Mathematica file that I used today. It is called ColoredGraph.nb. Right-click on the underlined word "here"; in the pop-up menu that appears your browser will offer you to save the file in your directory. Make sure that the file is saved with exactly this name: ColoredGraph.nb. After saving the file you can open it from Mathematica. Start Mathematica, then click on the menu item File, choose Open, and then find ColoredGraph.nb. When you open this file place the cursor in the first cell and press Shift+Enter. Then continue with other cells. You can get all the images which are presented above.
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There are remarkable webpages that deal with visualizations of complex functions. Below I list only two. The first page gives a long list of additional resources. The second provides a Mathematica package to do plots similar to ones above.
- Hans Lundmark's complex analysis page.
- Bernd Thaller's VQM packages for visualizing much more than complex functions; see his examples here.
- Suggested problems for Section 2.1: 1, 3, 6, 7, 12-16, 17, 29, 21, 24-26, 27-29, 31, 33-36.
- Suggested problems for Section 2.2: 1, 4, 6, 8, 9-14, 15, 16, 18-20, 21-26, 27, 28, 30-32.
- Suggested problems for Section 2.3: 4-6, 11, 12, 15, 16, 17, 18, 20, 21, 24-26, 28, 29, 33, 35-37.
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Below it the unit circle with the most important angles and their cosines and sines. Notice that the unit circle is the set of all complex numbers $z =x + i y$ such that $|z| =1.$ The picture below can be viewed as a "cheat sheet" for the conversion of some important complex numbers with modulus $1$ from the standard form (that is the form $z =\operatorname{Re} z + i\,\operatorname{Im} z$) to the polar form and vice versa.
Click on the image to get a pdf file.
- The above image is a modified picture that I found at the Wikipedia page about the unit circle. If you compare the picture on the Wikipedia page you will notice that I labeled the angles with the values of the principal argument, that is the values from the interval $(-\pi,\pi].$ I also removed the degree values, since degrees are not used in complex analysis.
- See also the page on roots of unity.
- Suggested problems for Section 1.4: 1-17, 19, 20, 21, 24, 25, 29, 31, 34
- Suggested problems for Section 1.5: 1-12, 13-24, 25, 27, 28, 29, 33, 37, 38
- Here is the Wikipedia article about: $e^{i t} = \cos t + i \sin t$ (Euler's formula). Please read the whole article, or read the parts that you find interesting. The proof that I presented in class is one of three proofs presented at the bottom of the page.
- Today in class we encountered the Pythagorean theorem. You must have seen a proof of this theorem in other classes. But, there are many, many proofs of this theorem; as often Wikipedia is a good source of information. If you scroll to September 29, 2009 on this page you can find a visual proof of the Pythagorean theorem. You might not seen such a proof before.
- I did prove (and you should know this proof) the triangle inequality in class:
- For all complex numbers $z$ and $w$ we have: $|z+w| \leq |z|+|w|.$
- The information sheet
- Suggested problems for Section 1.1: 1, 2(a)(c), 4, 6, 7, 9, 10, 12, 14, 15, 16, 18, 21, 24, 25, 27, 28, 30-33, 35, 36, 38, 39, 40, 45, 46, 50
- Suggested problems for Section 1.2: 1, 4, 6, 7, 8, 9, 12, 14, 15, 19, 20, 21, 22, 23, 26-29, 31, 33, 34, 36, 37, 39, 41
- Suggested problems for Section 1.3: 1-12, 15, 17-20, 22, 23, 25-27, 33-36, 39-44
- A related Wikipedia link: Complex number.