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Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremefif1o 20401* The exponential function of an imaginary number maps any open-below, closed-above interval of length one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.)

Theoremefifo 20402* The exponential function of an imaginary number maps the reals onto the unit circle. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremeff1olem 20403* The exponential function maps the set , of complex numbers with imaginary part in a real interval of length , one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.)

Theoremeff1o 20404 The exponential function maps the set , of complex numbers with imaginary part in the closed-above, open-below interval from to one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)

13.3.4  The natural logarithm on complex numbers

Syntaxclog 20405 Extend class notation with the natural logarithm function on complex numbers.

Syntaxccxp 20406 Extend class notation with the complex power function.

Definitiondf-log 20407 Define the natural logarithm function on complex numbers. See http://en.wikipedia.org/wiki/Natural_logarithm ("The natural logarithm function can also be defined as the inverse function of the exponential function"). (Contributed by Paul Chapman, 21-Apr-2008.)

Definitiondf-cxp 20408* Define the power function on complex numbers. Note that the value of this function when and should properly be undefined, but defining it by convention this way simplifies the domain. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremlogrn 20409 The range of the natural logarithm function, also the principal domain of the exponential function. This allows us to write the longer class expression as simply . (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)

Theoremellogrn 20410 Write out the property explicitly. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremdflog2 20411 The natural logarithm function in terms of the exponential function restricted to its principal domain. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremrelogrn 20412 The range of the natural logarithm function includes the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 1-Apr-2015.)

Theoremlogrncn 20413 The range of the natural logarithm function is a subset of the complex numbers. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremeff1o2 20414 The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)

Theoremlogf1o 20415 The natural logarithm function maps the nonzero complex numbers one-to-one onto its range. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremdfrelog 20416 The natural logarithm function on the positive reals in terms of the real exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremrelogf1o 20417 The natural logarithm function maps the positive reals one-to-one onto the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremlogrncl 20418 Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremlogcl 20419 Closure of the natural logarithm function. (Contributed by NM, 21-Apr-2008.) (Revised by Mario Carneiro, 23-Sep-2014.)

Theoremlogimcl 20420 Closure of the imaginary part of the logarithm function. (Contributed by Mario Carneiro, 23-Sep-2014.) (Revised by Mario Carneiro, 1-Apr-2015.)

Theoremlogcld 20421 The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 20419. (Contributed by David Moews, 28-Feb-2017.)

Theoremlogimcld 20422 The imaginary part of the logarithm is in . Deduction form of logimcl 20420. Compare logimclad 20423. (Contributed by David Moews, 28-Feb-2017.)

Theoremlogimclad 20423 The imaginary part of the logarithm is in . Alternate form of logimcld 20422. (Contributed by David Moews, 28-Feb-2017.)

Theoremabslogimle 20424 The imaginary part of the logarithm function has absolute value less than pi. (Contributed by Mario Carneiro, 3-Jul-2017.)

Theoremlogrnaddcl 20425 The range of the natural logarithm is closed under addition with reals. (Contributed by Mario Carneiro, 3-Apr-2015.)

Theoremrelogcl 20426 Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremeflog 20427 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremreeflog 20428 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremlogef 20429 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremrelogef 20430 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremlogeftb 20431 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremrelogeftb 20432 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremlog1 20433 The natural logarithm of . One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremloge 20434 The natural logarithm of . One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremlogneg 20435 The natural logarithm of a negative real number. (Contributed by Mario Carneiro, 13-May-2014.) (Revised by Mario Carneiro, 3-Apr-2015.)

Theoremlogm1 20436 The natural logarithm of negative . (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)

Theoremlognegb 20437 If a number has imaginary part equal to , then it is on the negative real axis and vice-versa. (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremrelogoprlem 20438 Lemma for relogmul 20439 and relogdiv 20440. Remark of [Cohen] p. 301 ("The proof of Property 3 is quite similar to the proof given for Property 2"). (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremrelogmul 20439 The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremrelogdiv 20440 The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremexplog 20441 Exponentiation of a nonzero complex number to an integer power. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremreexplog 20442 Exponentiation of a positive real number to an integer power. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremrelogexp 20443 The natural logarithm of positive raised to an integer power. Property 4 of [Cohen] p. 301-302, restricted to natural logarithms and integer powers . (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremrelog 20444 Real part of a logarithm. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremrelogiso 20445 The natural logarithm function on positive reals determines an isomorphism from the positive reals onto the reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremreloggim 20446 The natural logarithm is a group isomorphism from the group of positive reals under multiplication to the group of reals under addition. (Contributed by Mario Carneiro, 21-Jun-2015.)
flds        mulGrpflds        GrpIso

Theoremlogltb 20447 The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremlogfac 20448* The logarithm of a factorial can be expressed as a finite sum of logs. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremeflogeq 20449* Solve an equation involving an exponential. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremlogne0 20450 Logarithm of a non-1 number is not zero and thus suitable as a divisor. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremlogleb 20451 Natural logarithm preserves . (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremrplogcl 20452 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 21-Sep-2014.)

Theoremlogge0 20453 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremlogcj 20454 The natural logarithm distributes under conjugation away from the branch cut. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremefiarg 20455 The exponential of the "arg" function . (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremcosargd 20456 The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 20455. (Contributed by David Moews, 28-Feb-2017.)

Theoremcosarg0d 20457 The cosine of the argument is zero precisely on the imaginary axis. (Contributed by David Moews, 28-Feb-2017.)

Theoremargregt0 20458 Closure of the argument of a complex number with positive real part. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremargrege0 20459 Closure of the argument of a complex number with nonnegative real part. (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremargimgt0 20460 Closure of the argument of a complex number with positive imaginary part. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremargimlt0 20461 Closure of the argument of a complex number with negative imaginary part. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremlogimul 20462 Multiplying a number by increases the logarithm of the number by . (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremlogneg2 20463 The logarithm of the negative of a number with positive imaginary part is less than the original. (Compare logneg 20435.) (Contributed by Mario Carneiro, 3-Apr-2015.)

Theoremlogmul2 20464 Generalization of relogmul 20439 to a complex left argument. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremlogdiv2 20465 Generalization of relogdiv 20440 to a complex left argument. (Contributed by Mario Carneiro, 8-Jul-2017.)

Theoremabslogle 20466 Bound on the magnitude of the complex logarithm function. (Contributed by Mario Carneiro, 3-Jul-2017.)

Theoremtanarg 20467 The basic relation between the "arg" function and the arctangent. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremlogdivlti 20468 The function is strictly decreasing on the reals greater than . (Contributed by Mario Carneiro, 14-Mar-2014.)

Theoremlogdivlt 20469 The function is strictly decreasing on the reals greater than . (Contributed by Mario Carneiro, 14-Mar-2014.)

Theoremlogdivle 20470 The function is strictly decreasing on the reals greater than . (Contributed by Mario Carneiro, 3-May-2016.)

Theoremrelogcld 20471 Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreeflogd 20472 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrelogmuld 20473 The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrelogdivd 20474 The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremlogled 20475 Natural logarithm preserves . (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrelogefd 20476 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrplogcld 20477 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremlogge0d 20478 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremdivlogrlim 20479 The inverse logarithm function converges to zero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlogno1 20480 The logarithm function is not eventually bounded. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 30-May-2016.)

Theoremdvrelog 20481 The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremrelogcn 20482 The real logarithm function is continuous. (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremellogdm 20483 Elementhood in the "continuous domain" of the complex logarithm. (Contributed by Mario Carneiro, 18-Feb-2015.)

Theoremlogdmn0 20484 A number in the continuous domain of is nonzero. (Contributed by Mario Carneiro, 18-Feb-2015.)

Theoremlogdmnrp 20485 A number in the continuous domain of is not a strictly negative number. (Contributed by Mario Carneiro, 18-Feb-2015.)

Theoremlogdmss 20486 The continuity domain of is a subset of the regular domain of . (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremlogcnlem2 20487 Lemma for logcn 20491. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremlogcnlem3 20488 Lemma for logcn 20491. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremlogcnlem4 20489 Lemma for logcn 20491. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremlogcnlem5 20490* Lemma for logcn 20491. (Contributed by Mario Carneiro, 18-Feb-2015.)

Theoremlogcn 20491 The logarithm function is continuous away from the branch cut at negative reals. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremdvloglem 20492 Lemma for dvlog 20495. (Contributed by Mario Carneiro, 24-Feb-2015.)
fld

Theoremlogdmopn 20493 The "continuous domain" of is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
fld

Theoremlogf1o2 20494 The logarithm maps its continuous domain bijectively onto the set of numbers with imaginary part . The negative reals are mapped to the numbers with imaginary part equal to . (Contributed by Mario Carneiro, 2-May-2015.)

Theoremdvlog 20495* The derivative of the complex logarithm function. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremdvlog2lem 20496 Lemma for dvlog2 20497. (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremdvlog2 20497* The derivative of the complex logarithm function on the open unit ball centered at , a sometimes easier region to work with than the of dvlog 20495. (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremadvlog 20498 The antiderivative of the logarithm. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremadvlogexp 20499* The antiderivative of a power of the logarithm. (Set and multiply by to get the antiderivative of itself.) (Contributed by Mario Carneiro, 22-May-2016.)

Theoremefopnlem1 20500 Lemma for efopn 20502. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

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