Complex analysis Definition and 784 Threads

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions).

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  1. S

    Entire Functions Bounded by Exponential Growth

    Homework Statement Find all entire functions f such that |f(z)|\leq e^{\textrm{Re}(z)}\quad\forall z\in\mathbb{C} Homework Equations \textrm{Re}(u+iv)=u The Attempt at a Solution I tried using Nachbin's theorem for functions of exponential type. I also tried using the Cauchy...
  2. D

    Is (-i)^(-m) Equal to cos((m*pi)/2)+i*sin((m*pi)/2) in Complex Analysis?

    is this relashion true? or false? if it is true how can I proof it? (-i)^(-m) = cos((m*pi)/2)+i*sin((m*pi)/2)
  3. J

    Complex analysis - graphing in complex plane

    Homework Statement Graph the following in the complex plane {zϵC: (6+i)z + (6-i)zbar + 5 = 0} Homework Equations z=x+iy zbar=x-iy The Attempt at a Solution Substituting the equations gives 2(6x-y) + 5 = 0 => y = 6x + (5/2) But that's a line in R^2. The imaginary parts...
  4. A

    Complex analysis (conformal?) mapping question probably easy

    Homework Statement We're supposed to find a bijective mapping from the open unit disk \{z : |z| < 1\} to the sector \{z: z = re^{i \theta}, r > 0, -\pi/4 < \theta < \pi/4 \}.Homework Equations The Attempt at a Solution This is confusing me. I tried to find a function that would map [0,1), which...
  5. A

    Complex analysis / Using analyticity of f to prove f is constant

    Homework Statement I'm supposed to show that, if f is analytic and |f| is constant on a domain D \subset \mathbb{C}, f is constant. Homework Equations The hint is to write f^* = |f|^2 / f. I might also need to use the fact that if f^* is analytic too, then f is constant. The Attempt...
  6. S

    Bilinear Maps Complex Analysis

    Homework Statement Find a function g analytic in |z|\leq 2, with g(2/3)=0 and |g(z)|= 1 on |z|=2 Homework Equations Bilinear maps B_{\alpha}(z)=\frac{z-\alpha}{1-\overline{\alpha}z} |B_{\alpha}(z)|=1 on |z|=1 The Attempt at a Solution I tried using the maximum...
  7. S

    Please very my solution: complex analysis

    Homework Statement 1) consider az - b*conj(z) + c = 0 where a,b,c are complex unknown constans express z in terms of a,b,c Homework Equations The Attempt at a Solutionok so i took the conjugate of the original equation to get a second equation: a*conj(z) - b*z + c = 0 so my two...
  8. S

    Analyzing Complex Functions with a Given Inequality - Homework Problem

    Homework Statement suppose that f(z) is an analytic function on all of C, and suppose that, for all z in C, we have |f(z)| <= sqrt{|z|} Homework Equations The Attempt at a Solution I'm unsure of how to start the proof. any help is greatly appreciated.
  9. L

    Complex Analysis Qn: Show Constant Function in B(z0; r)

    Homework Statement Suppose z0 = x0 + iy0 2 C, and r > 0. Further, suppose that f(z) is a real valued function that is analytic on the open box B(z0; r) = { x + iy | x0 < x < x0 + r; y0 < y < y0 + r }. Then show that f(z) must, in fact, be constant on the box B(z0; r). The Attempt at...
  10. D

    Proving |z|<1 and n is a positive integer: Complex Analysis Proof

    Homework Statement Given |z|<1 and n a positive integer prove that \left|\frac{1-z^n}{1-z}\right|\le n The Attempt at a Solution I try to find the maximum of the function by differentiation \frac{d}{dz}\frac{1-z^n}{1-z}=\frac{-nz^{n-1}*(1-z)+(1-z^n)}{(1-z)^2}=0\Rightarrow...
  11. N

    Complex Analysis: Inverse Laplace Transform

    Homework Statement Hi all. I have found the Laplace transform of the following piecewise function: f(x) = \left\{ {\begin{array}{*{20}c} {0\,\,\,\,{\rm{for}}\,\,\,\,x < 0} \\ {x\,\,\,\,{\rm{for}}\,\,x \in (0;1)} \\ {0\,\,\,\,{\rm{for}}\,\,\,\,x > 1} \\ \end{array}} \right. I...
  12. N

    Contribution of Semi-Circle in Complex Analysis Integration

    Homework Statement Hi all. I have the following integral: I = \int_{2 - i\infty}^{2+i\infty}{f(s) \exp(st)ds}, where f(s) is some function. In order to perform this integral, I will choose to close the vertical line with a semi-circle in some halfplane (in order to use Cauchy's integral...
  13. N

    Analyzing Singularities in Complex Functions

    Hi all We look at f(z)=\sqrt z . Here the point z0=0 is a branch point, but can/is z0=0 also regarded as a zero?
  14. N

    Can the Complex Plane Extend to Infinity?

    Homework Statement Hi all. We we look at z\rightarrow \infty, does this include both z=x for x \rightarrow \infty AND z=iy for y\rightarrow \infty? So, I guess what I am asking is, when z\rightarrow \infty, am I allowed to go to infinity from both the real and imaginary axis? If yes, then this...
  15. S

    Rudin's real and complex analysis solutions

    Hey, I'm studying Rudin's Real and Complex Analysis by myself and it would be really nice if I could find a solution manual to all/part of the exercises at the end of the chapters. Does anyone know if such a solution manual exists? Thanks
  16. D

    Complex analysis harmonic function

    I seem to be missing a subtlety of the definition of a harmonic function. I'm using Churchill and Brown. As stated in the book, an analytic function in domain D with component functions (i.e. real and imaginary parts) u(x,y) and v(x,y) are harmonic in D. harmonic functions satisfy uxx+uyy=0...
  17. D

    Showing Uniqueness of z^(1/3), z^(1/2) & ln(z) in Complex Plane

    How does one show that z^{1/3} is not unique in the complex plane? [ Similarly for z^(1/2) and ln(Z) ] Thanks, Daniel
  18. N

    Complex Analysis: Poles and Singularities

    Homework Statement Hi all. According to my book, a pole z_0 of a function f(z) is defined as \mathop {\lim }\limits_{z \to z_0 } f(z) = \infty. Now let's look at e.g. f(z) = exp(z). Thus we have a singularity for z = infinity, since the limit in this case is infinity. This is what I don't...
  19. N

    Complex Analysis Q&A - Singularities, Integration and More

    Hi all. I have some questions on complex analysis. They are very fundemental. 1) Singularities of a complex functions are the points, where the functions fails to be analytic. Will a singularity then always be a point, where the numerator of the functions is zero? 2) This question is on...
  20. N

    Complex Analysis: Integrating rational functions

    Homework Statement Hi all. My question has to do with integrating rational functions over the unit circle. My example is taken from here (page 2-3): http://www.maths.mq.edu.au/%7Ewchen/lnicafolder/ica11.pdf We wish to integrate the following \int_0^{2\pi } {\frac{{d\theta }}{{a + \cos...
  21. Q

    Schwarz's lemma, complex analysis proof

    Homework Statement Let B1 = {z element C : abs(z) < 1}, f be a holomorphic function on B1 with Re f(z) > greater than or equal to 0 and f(0) =1. then show that: abs(f(z)) less than or equal to [(1+abs(z))/(1-abs(z))] Homework Equations Schwarz's Lemma: Suppose that f...
  22. Q

    Complex analysis, finding a bijection

    Homework Statement Let Omega = C\((-inf,-1]U[1,inf)), find a holomorphic bijection phi:omega-->delta, where delta is the open unit disk Homework Equations Reimann Mapping Theorem Special Mapping formulas: can map wedges onto wedges, with deletion of real line from zero to infinity in...
  23. J

    Usefulness of complex analysis for the physical sciences?

    Hi, I am interested in taking a complex analysis course. How useful is it to the physical sciences?
  24. S

    Argument Theorem - Complex Analysis

    Homework Statement Evaluate (1/2ipi)* contour integral of [z^(n-1)] / [(3z^n) - 1 ] dz Homework Equations I would assume you would have to use the Argument Theorem since this problem comes after the proof of the argument theorem in my book. The Attempt at a Solution z^(n-1)...
  25. G

    Proving the Area Enclosed by a Simple Closed Path using Complex Analysis

    Homework Statement C = positively oriented simple closed piecewise smooth path Prove that: (1/2i)*\int_{C}\bar{z}dz is the area enclosed by C. Homework Equations *I know that the curve C is piecewise smooth so that it can be broken up into finitely many pieces so that each piece...
  26. S

    Complex analysis: laurent, residues

    This is addressed to people who know complex analysis (hope this is the right section). Here's the Laurent theorem from my book for my later reference: Suppose a function f is analytic throughout an annular domain R1<|z-z0|<R2, centered at z0, and let C denote any positively oriented simple...
  27. P

    Is an Entire Function Satisfying f(z+i)=f(z) and f(z+1)=f(z) Constant?

    Homework Statement if an entire function satisfies f(z+i)=f(z) and f(z+1)=f(z), must the function be constant? Homework Equations The Attempt at a Solution It's true that f(0) = f(k) = f(ik) where k is an integer. I'm wondering whether I can apply Liouville's theorem into this...
  28. O

    Prove No Analytic Function F on Annulus D: 1<|z|<2

    Homework Statement Prove that there does not exist an analytic function on the annulus D: 1<|z|<2, s.t. F'(z) = 1/z for all z in D. Homework Equations The Attempt at a Solution Assume F exists, then for z in D, not a negative number, F(z) = Log z + c since Log' z = 1/z... Lost
  29. D

    Complex analysis definite integral involving cosine

    Homework Statement integral 1/(a+cos(t))^2 from 0 to pi. Homework Equations cos(t)=1/2(e^it+e^-it) z=e^it dz/(ie^it)=dt The Attempt at a Solution int dt/(a+cos(t))^2 = int dz/iz(a2+az+az-1+z2/4 +1/2 +z-2/4) so with these types of problems I normally can factor this guy...
  30. O

    Analytic Function Mapping to a Line: Constant Throughout Domain?

    Homework Statement Show that if the analytic function w= f(z) maps a domain D onto a portion of a line, then f must be constant throughout D. Homework Equations The Attempt at a Solution I just have one question, can I write w = u(x,y) (a+bi) since it maps to a portion of a line...
  31. L

    Complex analysis - multivalued functions

    What are the implications for holomorphicity of a function being a multifunction. take f(z)=\ln{z}=\ln{r}+i arg(z), here z=z_0+2k \pi all correspond to the same value of z but give different values of f(z) i.e. its a multifunction. how does this affect its holomorphicity? as far as i...
  32. I

    Complex Analysis - Removing A Singularity

    Ok, so I'm suppose to be able to remove the singularity to find the residue of the function (z)cos{\frac{1}{z} I tried to see how "bad" the singularity was by taking the limit, but I can't figure out if \lim_{ z \to 0 } (z)cos{\frac{1}{z} goes to 0 or if it is...
  33. F

    Need Recommendations for Books on Complex Analysis and Electrical Networks?

    Hey, I am looking for a good book on complex analyis (complex calculus, "complexe anlysis" in german). Any recommendations? I am a first year Electrical engineering student at the ETH Zürich. It should cover the following, and have a reasonalbe amount of examples: Analytical Funktions...
  34. A

    Complex Analysis: Analytic Function F(z)?

    Homework Statement Hey guys. I have this question, I took it from a test. I need to check if there is an analytic function F(z) in this area (in the pic) that has this derivative (in the pic). http://img256.imageshack.us/img256/7826/25453238.jpg Well, the derivative is analytic in...
  35. W

    Uniqueness of Holomorphic Functions

    Homework Statement Let f and g be two holomorphic functions in a connected open set D of the plane which have no zeros in D; if there is a sequence an of points such that lim an = a and an does not equal a for all n, and if f'(an)/f(an)=g'(an)/g(an) show that there is a constant c such that...
  36. L

    Complex Analysis Questions: Singularities and Integrals

    two questions here: (i) my notes say that \frac{1}{e^{\frac{1}{z}}-1} has an isolated singularity at z=\frac{1}{2 \pi i n}, n \in \mathbb{Z} \backslash \{0\} i can't see this though... (ii) let b \in \mathbb{R}. show \int_{-\infty}^{\infty} e^{-x^2} \cos{(2bx)} dx = e^{-b^2}...
  37. L

    Complex Analysis: Integrate e^{\sin{z}} \cos{z} over Curve w_1 to w_2

    Let w_1,w_2 \in \mathbb{C} and \gamma be some smooth curve from w_1 to w_2. Find \int_{\gamma} e^{\sin{z}} \cos{z} dz this is holomorphic on the entire copmlex plan so we can't use a residue theorem. furthermore, we can't assume \gamma is a closed contour as we aren't told w_1=w_2 so it...
  38. C

    Adv. Math for Engineers and Scientists or App. Complex Analysis?

    I'm a physics major and I have space for one more class the coming fall semester: either advanced mathematics for engineers and scientists or applied complex analysis. Advanced Mathematics for Engineers and Scientists- Vector analysis, Fourier analysis and partial differential equations...
  39. MathematicalPhysicist

    A question from Real and Complex Analysis (Rudin's).

    I am trying to understand theorem 1.17 in page 15-16 international edition 1987. How do you show that \phi_n(t) is a monotonic increasing sequence of functions?
  40. T

    Applications of Complex Analysis in Quantum Physics?

    Hi, I just finished up a Complex Analysis course last term and, though I'm no physics major, I thought Quantum Physics looked interesting. Does anyone know some common or interesting applications of Complex Analysis within Quantum Physics? Or even an online resource that might delve into...
  41. A

    How Can I Solve This Complex Integral Using Trigonometry or Complex Analysis?

    Homework Statement Hey guys. I have this integral, I tried to use trigo, tried to use the complex expression but nothing worked, can I please have some help? Thanks a lot. Homework Equations The Attempt at a Solution
  42. S

    Complex Analysis: Integration

    Homework Statement Evaluate the following integral for 0<r<1 by writing \cos\theta = \frac{1}{2}(e^{i\theta} + e^{-i\theta}) reducing the given integral to a complex integral over the unit circle. Evaluate: \displaystyle{\frac{1}{2\pi}\int_0^{2\pi}\frac{1}{1-2r\cos\theta +...
  43. A

    Complex Analysis: Solving Integral Problem with Sin

    Homework Statement Hey guys. So, I need to calculate this integral, I uploaded what I tried to do in the pic. But according to them, this is not the right answer, according to them, the right answer is the one I marked in red at the bottom. Any idea where this Sin came from? Thanks...
  44. A

    Complex Analysis Homework: Calculating Integral

    Homework Statement Hey guys. So, I need to calculate this integral. I uploaded what I tried to do. First of all, did the substitute, then I tried to use the residue theorem so I was looking for the residue of z=0 which is happen to be a removable singular point so it's just 0, then I...
  45. A

    Complex Analysis Homework: Need Help Showing Statement is True

    Homework Statement Hey guys. I have this problem, I need to show that it's true and I don't have a clue. I tried to do like alpha = x+yi but it got me nowhere, any ideas? Thanks. Homework Equations The Attempt at a Solution
  46. B

    Complex Analysis 2nd Ed. by Stephen D. Fisher: Q&A

    Hi, I'm studying complex analysis right now, I would like to use this thread to ask questions when I read books. Many questions will be very stupid, so please bear with me. Also, English is my second language. text: Complex Analysis (2nd edition) author: Stephen D. Fisher [question deleted]...
  47. G

    BME- Mechanics, Complex Analysis, Thermodynamics, Quantum

    Hi everyone, I'm a biochemistry major hoping to go into BME (ideally PhD). Besides taking a bunch of extra math courses, I made a list of engineering and intermediate level physics classes that grad schools seem to be looking for, and also kind of figured out what courses offered by my school...
  48. Q

    Complex Analysis Proof showing that a Polynomial is linear

    Homework Statement Suppose P is a polynomial such that P(z) is real iff. z is real. Prove that P is linear. The hint given in the text is to set P = u + iv, z = x+iy and note that v = 0 iff y = 0. We are then told to conclude that a. either v-sub y(partial of v with respect to y) is...
  49. S

    Complex Analysis: Proving Vector z1 Parallel to z2

    Homework Statement Show that the vector z1 is parallel to z2 if and only if Im(z1z2*)=0 note: z2* is the complement of z2 Homework Equations The Attempt at a Solution I would probably convert z to polar form. so, z1=r1(cos Ѳ1+isin Ѳ1) z2=r2(cos Ѳ2+isin Ѳ2) so...
  50. K

    Complex Analysis Graphing Question

    Homework Statement I want to show z_{1}+(z_{2}+z_{3})=(z_{1}+z_{2})+z_{3} with the use of a graph. Homework Equations The Attempt at a Solution I am just cluless on how to graph. I know z=x+iy where the real part is on the x-axis and the imaginary part is on the y axis.
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