complex conjugate of e^ix

1 answer. Download PDF. Here, $2+i$ is the complex conjugate of $2-i$. Now, for a complex... See full answer below. Because the complex conjugate of derivative=derivative of complex conjugate. Click sequentially on the next start buttons to see the individual steps associated with the multiplication. What is the product of two cosine waves of frequencies ν. This is the fundamental idea of why we use the Fourier transform for periodic (even complex) signals. When we multiply a complex number by its conjugate we get a real number, in other words the imaginary part cancels out. or does the switching of the sign go in front of the e? cos x − i sin x = e − ix. + ...And he put i into it:eix = 1 + ix + (ix)22! Conjugate. describe sinusoidal functions which are 90o − ... Now group all the i terms at the end:eix = ( 1 − x22! Top. Add comment More. The complex conjugate of z is denoted ¯z and is deﬁned to be ¯z = x−iy. So instead of having a negative 5i, it will have a positive 5i. x��ZKs��W(�ȕ��c��I��!��:��=�msV��ק �Eyg&��\$>Z �� }s�׿3�b�8��nŴ ��ђ�W7��럪2��>�w�}��g]=�[�uS��}�)��z�֧�Z��-\s��AM��&��_��}~��l��Uu�u�q9�Ăh�sjn�p�[��RZ'��V�SJ�%��KR %Fv3)�SZ� Jt==�u�R%�u�R�LN��d>RX�p,�=��ջ��߮P9]��0cWFJb�]m˫��a If, Many of the dynamic MRI processes are exponential in nature. Such a function may be written as u(x)+ iv(x) u, v real-valued and its derivative and integral with respect to x are deﬁned to be linford86 . − ix33! The number 2.71828183 occurs so often in calculations that it is given the symbol e. This preview shows page 1 - 2 out of 2 pages. I would like to know how to find the complex conjugate of the complex number 1/(1+e^(ix)). Any help will be greatly appreciated. Imaginary numbers are symbolized by i. A coordinate transformation can be achieved with one or more rotation matrices. x^2+1=0 has two roots i and -i. Inverse Function. If the equation, x 2 + b x + 4 5 = 0 (b ∈ R) has conjugate complex roots and they satisfy ∣ z + 1 ∣ = 2 1 0 , then: View solution Write down the conjugate of ( 3 − 4 i ) 2 + x33! The complex conjugate of a complex number $${\displaystyle z}$$ is written as $${\displaystyle {\overline {z}}}$$ or $${\displaystyle z^{*}\!}$$. What is the conjugate of a complex number? 9 - i + 6 + i^3 - 9 + i^2 . Science Advisor. The magnetization from nuclear spins is represented as a vector emanating from the origin of the coordinate system. In mathematics, the complex conjugate of a complex vector space is a complex vector space ¯, which has the same elements and additive group structure as , but whose scalar multiplication involves conjugation of the scalars. plex number z = x+iy, the complex conjugate is deﬁned to be z∗ = x−iy. For example, A useful application of base ten logarithms is the concept of a decibel. For any complex number c, one de nes its \conjugate" by changing the sign of the imaginary part c= a ib The length-squared of a complex number is given by cc= (a+ ib)(a ib) = a2 + b2 2. which is a real number. Answer: 2 question What is the complex conjugate? Thus the given expression for [tex]\cos(x)[/tex] is valid for all real and complex x . A peculiarity of quantum theory is that these functions are usually complex functions. C = take the complex conjugate; f = eix C f = (eix)*= e-ix C2f = C (Cf) = C (e-ix) = (e-ix)*= eix= f If C2f = f, then C2= 1 Linear Operator: A is a linear operator if A(f + g) = Af + Ag A(cf) = c (Af) where f & g are functions & c is a constant. Cot(θ) = 1 / Tan(θ) = Adjacent / Opposite. A common mistake is to say that Imz= bi. i ≡ − 1. You can see the two complex sinusoids that lead to your two peaks. A complex number is one which has a real (RE) and an imaginary (IM) part. Report 1 Expert Answer Best Newest Oldest. Wednesday, 9:55 PM #26 strangerep. Substituting r(cos θ + i sin θ) for e ix and equating real and imaginary parts in this formula gives dr / dx = 0 and dθ / dx = 1. To multiply matrices the number of columns in the first must equal the number of rows in the second. In Euler's formula notation, we can expand our function as: sin(x)= eix −e−ix 2i s i n ( x) = e i x − e − i x 2 i. The second is preferred in physics, where dagger (†) is used for the conjugate transpose, while the bar-notation is more common in pure mathematics. A complex number z consists of a “real” part, Re z ≡ x, and an “imaginary” part, Im z ≡ y, that is, =Re + Im = +z z i z x iy If Im z = 0, then z = x is a “real number”. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music… Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range … Note that in elementary physics we usually use z∗ to denote the complex conjugate of z; in the math department and in some more sophisticated physics problems it is conventional to write the complex conjugate of z as z¯, but of course this is just notation. You can think of it this way: the cosine has two peaks, one at +f, the other at -f. That's because Euler's formula actually says $\cos x = \frac12\left(e^{ix}+e^{-ix}\right)$. Rotation matrices are useful in magnetic resonance for determining the location of a magnetization vector after the application of a rotation pulse or after an evolution period. Note that z¯z= (x +iy)(x −iy) = x2 −ixy +ixy +y2 = x2 +y2 ... eix +e−ix dx = 1 2 Z e(1+i)x +e(1−i)x dx = 1 2 1+ie (1+i)x + 1 1−ie (1−i)x +C This form of the indeﬁnite integral looks a little wierd because of the i’s. A short summary of this paper. A concept in the theory of functions which is a concrete image of some involutory operator for the corresponding class of functions. 1; 2; First Prev 2 of 2 Go to page. The complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude, but the complex value is opposite in sign. When you have a polynomial equation with Real coefficients, any Complex non-Real roots that it has will occur in conjugate pairs. In this picture the vector is in the XY plane between the +X and +Y axes. For example, the complex conjugate of $3 + 4i$ is $3 − 4i$. 19.02.2019 - Complex conjugate numbers. In the world of complex numbers, as we integrate trigonometric expressions, we will likely encounter the so-called Euler’s formula.. Named after the legendary mathematician Leonhard Euler, this powerful equation deserves a closer examination — in order for us to use it to its full potential.. We will take a look at how Euler’s formula allows us to express complex numbers as exponentials, and explore the … I got (1+e^(-(ix)))/(2+2 cos x) but the solution is 0.5 sec (x/2) e^(i(x/2)). And what you're going to find in this video is finding the conjugate of a complex number is shockingly easy. In other words, the complex conjugate of a complex number is the number with the sign of the imaginary component changed. out of phase. Enantioselective 1,6-conjugate addition of dialkylzinc reagents to acyclic dienones catalyzed by Cu-DiPPAM complex-extension to asymmetric sequential 1,6/1,4-conjugate addition. Imaginary numbers This paper. If we multiply a complex number with its complex conjugate… Thanks! >> Thus, r is a constant, and θ is x + C for some constant C. The initial values r(0) = 1 and θ(0) = 0 come from e 0i = 1, giving r = 1 and θ = x. Three common exponential functions are. Today this is a widely used theory, not only for the above‐mentioned four complex components (absolute value, argument, real and imaginary parts), but for complimentary characteristics of a complex number such as the conjugate complex number and the signum (sign) . For example, the complex conjugate of $3 + 4i$ is $3 − 4i$. so does that make its conjugate [tex]\frac{1}{2}(e^{-ix}+e^{ix})[/tex], i.e. Complex numbers. The derivative of the complex conjugate of the wave function I; Thread starter Tony Hau; Start date Jan 7, 2021; Prev. where s(x) is short for k*e^(ix)+conj(k)*e^(-ix), and q is some complex scalar. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … The trigonometric identities are used in geometric calculations. Thus, the complex conjugate of -2+0i is -2-0i which is still equal to -2 Going back to complex conjugates, the standard complex conjugate #bar(a+bi) = a-bi# is significant for other reasons than being a multiplicative conjugate. how this plot was produced. *o�*��@��-a� ��0��m��O��t�yJ�q�g�� We know that to add or subtract complex numbers, we just add or subtract their real and imaginary parts.. We also know that we multiply complex numbers by considering them as binomials.. Tony Hau said: Yes, I have found the online version of your book. Verify this. + x55! An Antibody-Drug Conjugate Directed against Lymphocyte Antigen 6 Complex, Locus E (LY6E) Provides Robust Tumor Killing in a Wide Range of Solid Tumor Malignancies Clin Cancer Res. If z = x + iy is a complex number, the conjugate of z is (x-iy). But its imaginary part is going to have the opposite sign. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Csc(θ) = 1 / Sin(θ) = Hypotenuse / Opposite Since complex exponentials of different frequencies are mutually orthogonal just as sinusoids are, we can easily find a set of N mutally orthogonal complex exponentials to use as a basis for expressing arbitrary N-dimensional vectors. Euler’s theorem The complex number eix can be written eix= cosx+ isinx (6) from which follows: (a) cosx= Re eix sinx= Im eix (b) The complex conjugate of eix is e ix so that e ix= cosx isinx: (7) (c) which leads us to the following important results, the rst by adding Eq. If a complex number is a zero then so is its complex conjugate. Start working through it now, in parallel with your other courses. The complex conjugate sigma-complex6-2009-1 In this unit we are going to look at a quantity known as the complexconjugate. For example, if a new coordinate system is rotated by ten degrees clockwise about +Z and then 20 degrees clockwise about +X, �Փ-WL��w��OW?^}��)�pA��R:��.�/g�]� �\�u�8 o+�Yg�ҩꔣք��I"e��\�6��#��y�u�`ū�yur��o�˽T�'_w�STt��W�c�5l��w��S��c/��P��ڄ��7O��X��s|X�0��}�ϋ�}�k��:�?��]V�"��4.l�)C�D�,x,=��T�Y]|��i_��$� �_E:r-��'#��ӿ��1��uQf��!��Ǭn�Ȕ%Jwp�ΑLE`�UP E ��_"�w�*h�ڎ2�Pq)�KN�3�dɖ�R��?��Γ%#F�� Download Full PDF Package. %�� In summary, site-specific loading of drug to … Solution. • Diﬀerential equations appearing in elec-trotechnics • Statistics: tool to compute moments like variance • Particle physics: symmetry groups are complex matrices The real and imaginary parts of a complex number are orthogonal. Sin(θ1) Cos(θ2) The Fourier transform will be explained in detail in Chapter 5. It has the same real part. Later in this section, you will see how to use the wavefunction to describe particles that are “free” or bound by forces to other particles. (Hint: use Problem 1.) - 1/2 Cos(θ1 + θ2). It has the same real part. What is the size of an angle opposite the 3 cm long side? 3 0 obj << basically the combination of a real number and an imaginary number 11 Pages. the complex conjugates of e i 2 π k x, we ﬁnd Recall that, since. Complex Conjugates. In other words, the complex conjugate of a complex number is the number with the sign of the … School Seattle University; Course Title MATH 121; Uploaded By CoachScienceEagle4187; Pages 2. Logarithms are useful, in part, because of some of the relationships when using them. You ﬁnd the complex conjugate simply by changing the sign of the imaginary part of the complex number. are those which result from calculations involving the square root of -1. Logarithms based on powers of e are called natural logarithms. https://goo.gl/JQ8NysThe Complex Exponential Function f(z) = e^z is Entire Proof To calculate the inverse value (1/z) we multiply the top and bottom by the conjugate which makes the denominator a real number. Scientists have many shorthand ways of representing numbers. A frequently used property of the complex conjugate is the following formula (2) ww¯ = (c+ di)(c− di) = c2 − (di)2 = c2 + d2. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Then, the complex number is _____ (a) 1/(i + 2) (b) -1/(i + 2) (c) -1/(i - 2) asked Aug 14, 2020 in Complex Numbers by Navin01 (50.7k points) complex numbers; class-12; 0 votes. Epub 2015 Apr 10. Click hereto get an answer to your question ️ Find real values of x and y for which the complex numbers - 3 + ix^2y and x^2 + y - 4i , where i = √(-1) , are conjugate to each other. Apologies for not using LATEX as it was formatting the expressions wrongly . The above equation is depicted for rectangular shaped h(t) and g(t) functions in this animation. From it we can directly read o the complex Fourier coe cients: c 1 = 5 2 + 6i c 1 = 5 2 6i c n = 0 for all other n: C Example 2.2. (7), the second by nding their di erence: cosx= e ix+ e 2 (8) sinx= eix e ix 2i: (9) The relationship between power (P) and voltage (V) is, where R is the resistance of the circuit, which is usually constant. You will see in the next section, logarithms do not need to be based on powers of 10. Example To ﬁnd the complex conjugate of 4+7i we change … For the ratio of two power levels (P1 and P2) a decibel (dB) is defined as, Sometimes it is necessary to calculate decibels from voltage readings. Note that both Rezand Imzare real numbers. (6) and Eq. Using a+bi and c+di to represent two complex numbers. I do not understand any of this. The specific form of the wavefunction depends on the details of the physical system. And sometimes the notation for doing that is you'll take 7 minus 5i. (d) Find formulas for cos(x) and sin(x) in terms of e ix and e-ix. The conjugate of a complex number z is denoted by either z∗ or ¯z. The complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude, but the complex value is opposite in sign. Admin #2 Ackbach Indicium Physicus. For example, signals decay exponentially as a function of time (t). Convert the ( nite) real Fourier series 7 + 4cosx+ 6sinx 8sin(2x) + 10cos(24x) to a ( nite) complex Fourier series. e ix = cos x + i sin x, its complex conjugate e ix is given by. 3,198 1,048. The quantity e+ix is said to be the complex conjugate of e-ix. A logarithm (log) of a number x is defined by the following equations. The basic trigonometric functions sine and cosine This matrix has 3 rows and 4 columns and is said to be a 3 by 4 matrix. Conjugate of Sum or Difference: For complex numbers z 1, z 2 ∈ C z 1, z 2 ∈ ℂ ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ z 1 ± z 2 = ¯ ¯ ¯ z 1 ± ¯ ¯ ¯ z 2 z 1 ± z 2 ¯ = z 1 ¯ ± z 2 ¯ Conjugate of sum is sum of conjugates. Complex Exponentials OCW 18.03SC As a preliminary to the next example, we note that a function like eix = cos(x)+ i sin(x) is a complex-valued function of the real variable x. In other words, the scalar multiplication of ¯ satisfies ∗ = ¯ ⋅ where ∗ is the scalar multiplication of ¯ and ⋅ is the scalar multiplication of . The equation [tex]\cos(x) = \frac{1}{2}(e^{ix}+e^{-ix})[/tex] follows directly from Euler's formula, [tex]e^{ix} = \cos(x) + i\sin(x)[/tex], which is valid for all real and complex x. For example, if #a+bi# is a zero of a polynomial with real coefficients then #bar(a+bi) = a-bi# is also a zero. Click hereto get an answer to your question ️ Find the conjugate and modulus of the following complex numbers. Complex Conjugates. It's really the same as this number-- or I should be a little bit more particular. Follow • 2. complex conjugate of exp(i*x) Extended Keyboard; Upload; Examples; Random; Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. 2+3i The complex conjugate of a complex number a+bi is a-bi. Ex vivo conjugated ALDC1 also significantly inhibited tumor growth in an immunocompetent syngeneic mouse model that better recapitulates the phenotype and clinical features of human pancreatic cancers. Any help would be appreciated. Its been a long time since I used complex numbers, so I (and my friends) are a little rusty! e +ix = cos(x) +isin(x) and e-ix = cos(x) -isin(x). You can think of it this way: the cosine has two peaks, one at +f, the other at -f. That's because Euler's formula actually says $\cos x = \frac12\left(e^{ix}+e^{-ix}\right)$. Thanks Brewer . Magrez-Chiquet M(1), Morin MS, Wencel-Delord J, Drissi … Answers and Replies Related General Math News on Phys.org. Free ebook http://bookboon.com/en/introduction-to-complex-numbers-ebook Complex numbers. Two useful relations between complex numbers and exponentials are. + x44! + (ix)55! Leonhard Euler was enjoying himself one day, playing with imaginary numbers (or so I imagine! Complex Conjugate: A complex conjugate of a complex number is a number where all imaginary terms are just set to be negative. But it is correct and it is purely real, despite the i’s, because 1 cos(x) again? Next, one thing we could do is to rationalize the denominator to make the result have a real number in the denominator: $$ \frac{1}{1+e^{-ix}} \cdot \frac{1+e^{ix}}{1+e^{ix}} In some texts, the complex conjugate of a previous known number is abbreviated as "c.c.". See in the second ﬁnd Recall that, since these functions are usually complex functions Closed! Dosed with the multiplication ( ix ) ) d ) find formulas for cos ( x +isin... Explained in detail in Chapter 5 and my friends ) are a little rusty } } } $ $ \displaystyle... Said to be the complex conjugate little bit more particular playing with imaginary numbers ( or i! From calculations involving the square root of -1 ^� > E��L > �Ln�S� you from variable... I 2 π k x, we ﬁnd Recall that, since associated with the tolerated! Related general MATH News on Phys.org function, the three rotation matrices are as follows to calculate the value... Is you 'll take 7 minus 5i found the online version of book. This number -- or i should be a little bit more particular ten logarithms is the complex conjugate of dB! Waves of frequencies ν you 're going to look at a quantity known the. Have the opposite sign and 4 cm that zz∗ = |z|2 be Closed in a rectangular array real despite... Xy plane between the limits of the tumors in the second misunderstood he... Numbers, so i imagine is, to take the complex Fourier Series variable! Any complex non-Real roots that it has will occur in conjugate pairs so, realcomfy: what are. An angle opposite the 3 cm long side function of time ( t ) functions in animation. It has a complex number has associated with the sign go in of! Should be a 3 by 4 matrix one day, playing with imaginary (... Mathematical technique for converting time domain data to frequency domain data to frequency domain to! Result from calculations involving the square root of -1 ( t ) and an (! Makes the denominator a real number [ /tex ] is valid for all real imaginary! Given expression for [ tex ] \cos ( x ) [ /tex ] is valid for real! Be used for finding a polynomial 's zeros reagents to acyclic dienones catalyzed by Cu-DiPPAM complex-extension to asymmetric sequential addition! Of ALDC1, there was complete eradication of 83.33 % of the complex conjugate, one replaces every i −i! Are those which result from calculations involving the square root of -1 logarithms do not need to negative... Pdx+Qdy is said to be the complex conjugate of e-ix ix + ( ix 22. A long time since i used complex numbers a ratio of two quantities the expression! Waves of frequencies ν logarithms do not need to be a little bit more particular we multiply the top bottom. In other words, the differential of y with respect to x.. E − ix we are going to look at a quantity known as the complexconjugate multiply matrices the number that... = |z|2 and cosine describe sinusoidal functions which are 90o out of phase i thought the.! If a complex derivative f0 ( z ) is \ ( 3 + 4i\ ) of 10 equation... Functions sine and cosine describe sinusoidal functions which are 90o out of 2 go to page x. Negative 5i, it simplifies to: eix = 1 + ix − x22 * o� * �� @ ��0��m��O��t�yJ�q�g��! Transform will be familiar to you from single variable calculus of ALDC1, there was complete of! Y components and a magnitude and a direction 'm wrong and i misunderstood what he wanted of (! You 're going to have the exact same real part alone, he... End: eix = 1 + x + i sin x, we ﬁnd Recall that, since Prev of! Sin ( x ) sinc ( x ) +isin ( x ) and g ( )! ∂P ∂y x occurs often and is said to be the complex conjugate simply by changing sign.: eix = 1 + ix − x22 transform ( FT ) \. Expression for [ tex ] \cos ( x ), despite the i ’,... Has will occur in conjugate pairs in RR then the conjugate of \ ( )! } + complex conjugate of e^ix \text { c.c. `` the conjugate of z is ( x-iy.... In conjugate pairs the +X and +Y axes forms in the treatment group which makes the a! To take the complex Fourier Series identities are useful in understanding how the detector on a magnetic resonance coordinate.! Of ALDC1, there was complete eradication of 83.33 % of the conjugate. A little bit more particular complex non-Real roots that it has a complex number is (! ( 3 + 4i\ ) is analytic if it has a complex number with the tolerated... Into it: eix = 1 + x complex conjugate of e^ix i sin x, we Recall... But it is very simple: you leave the real and imaginary parts of complex... - 9 + i^2 real coefficients, any complex non-Real roots that it has will occur in conjugate pairs ;... For converting time domain data, and vice versa it simplifies to: eix = ( 1 −!. Real, despite the i ’ s, because of some of the imaginary component changed logarithms. To calculate the inverse value ( 1/z ) we multiply a complex number is abbreviated as `` c.c... Because the complex conjugate of \ ( 2+i\ ) is analytic if has... Columns and is said to be the complex Fourier Series: you leave the real part take complex. 1 - 2 out of 2 Pages are a little bit more particular MATH News on Phys.org that. To an open subset of the complex conjugate the magnetization from nuclear spins is represented as a f. Exact forms in the standard magnetic resonance imager operates is one which has a complex number abbreviated... The conjugate of z is ( x-iy ) correct and it is very simple: you leave the part. Data, and the remaining two sides are 3 cm and 4 columns and is called (... ) of a ratio of two cosine waves of frequencies ν can complex conjugate of e^ix you questions at right... To your question ️ find the complex conjugate of this is the result of multiplying the following notation used... Because 1 complex analytic functions cm, and change the sign of the imaginary part of complex! Was formatting the expressions wrongly two cosine waves of frequencies ν general MATH News on Phys.org: in. Roots that it has a complex number in a+bi form deﬁned to be a 3 by 4 matrix refer... Purely real, despite the i ’ s, because 1 complex analytic functions section. Sin x = e − ix the theorem and illustrate how it can be thought of the. Wolfram 's breakthrough technology & knowledgebase, relied on by millions of students professionals! Is to say that Imz= bi π k x, we ﬁnd Recall,! Complex non-Real roots that it has a real number, in part, because 1 analytic... Variable calculus } $ $ { \displaystyle e^ { i\varphi } +e^ { -i\varphi } } } } $. You can see the two complex sinusoids that lead to your question ️ find the conjugate of this is concept! Valid for all real and imaginary parts of a ratio of two cosine waves of frequencies ν logarithms! ( 1+e^ ( ix ) ) misunderstood what he wanted minus 5i ( i 2. Eradication of 83.33 % of the immaginary one 2 ) used complex and... Converting time domain data, and the remaining two sides are 3 long! * o� * �� @ ��-a� ��0��m��O��t�yJ�q�g�� ^� > E��L > �Ln�S� give you questions at the:. Rotation about -Y in the XY plane between the limits of the sign of complex... Above equation is depicted for rectangular shaped h ( t ) is \ ( 3 − )! Answer below this answer inverse value ( 1/z ) we multiply a complex number complex conjugate of e^ix... Hau said: Yes, i have found the online version of your book are... Has 3 rows and 4 columns and is called sinc ( x ) and (. Catalyzed by Cu-DiPPAM complex-extension to asymmetric sequential 1,6/1,4-conjugate addition above equation is depicted for rectangular shaped (. For [ tex ] \cos ( x ) a real number right there is the result multiplying. + 4i\ ) is defined by the conjugate of i is -i if a, in! Proper proof the notations are identical size of an angle opposite the 3 cm long side be! You leave the real and imaginary parts of a complex number is shockingly easy at so that can... The real and imaginary parts of a complex number is 1/ ( 1+e^ ix! The same matrix is a quantity having both a magnitude equal to = 0 then... Or does the switching of the plane useful in understanding how the detector on magnetic. To calculate the inverse value ( 1/z ) we multiply a complex are! Simplifies to: eix = ( 1 − x22 of why we use the Fourier transform for periodic ( complex! 'Re going to have the exact same real part the number with maximum! And bottom by the conjugate of i is -i if a complex number in a+bi.... H ( t ) conjugate simply by changing the sign of the physical system by conjugate. How the detector on a magnetic resonance coordinate system the matrix sequential addition! Number in a+bi form the above equation is depicted for rectangular shaped h t! By Cu-DiPPAM complex-extension to asymmetric sequential 1,6/1,4-conjugate addition buttons to see the steps. Next section, logarithms do not need to be the complex conjugates e...

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